The Diffiety Institute Preprint Series

Preprint DIPS–3/2004 June 18, 2004

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Classes of the Maxwell spaces that admit subgroups of the Poincaré group

by

M. A. Parinov

Classes of the Maxwell spaces that admit subgroups of the Poincaré group

M. A. PARINOV

Abstract. The Maxwell space is the triple (M, g, F), where M is the four–dimensional Minkowski space or a domain in it, g is a pseudo-Euclidean metric on M, and F is a closed exterior 2-form on M. In this paper, we give the exhaustive description of classes of the Maxwell spaces that admit subgroups of the Poincaré group. Representatives of all classes are constructed.

Contents

1. Formulation of the problem and method of its solution 2 2. The classes of the Maxwell spaces that admit one-dimensional symmetry

groups 4 3. The Maxwell spaces that admit translation groups 14 4. The classes of the static Maxwell spaces 16 5. The Maxwell spaces that admit elliptic helices 20 6. The Maxwell spaces that admit hyperbolic helices 25 7. The Maxwell spaces that admit parabolic helices 32 8. The classes of the Maxwell spaces that admit proportional bi-rotations 43 9. Other classes 46 References 48

Introduction. In classical theory, the electromagnetic field is described by a skewsymmetric tensor F_{ij} on a 4-dimensional real manifold M ⊂ R_{1}^{4} (a domain in the Minkowski space) satisfying the Maxwell equations [1]

∂_{[i}F_{jk]} = 0, ∇_{k}F^{ik} = −\frac{4π}{c} J^{i} (i, j, k = 1, …, 4)

(the current J^{i} must satisfy the continuity equation ∇_{i}J^{i} = 0).

We shall say that the Maxwell space is a triple (M, g, F), where M is a smooth, real, four-dimentional manifold, F = \frac{1}{2}F_{ij}dx^{i} ∧ dx^{j} is a generalized symplectic structure on M, and g = g_{ij}dx^{i}dx^{j} is a pseudo-Euclidean metric on M of the Lorentz signature (− − −+).

2000 Mathematics Subject Classification. 83A05, 83C50, 53D99.

Key words and phrases. Minkowski space, Poincaré group, Maxwell equations, symplectic stucture, Maxwell space, classification.

Partially supported by the Ministry of Education of the Russian Federation grant 97-0-1.3-99.

This paper is submitted to the special Issue in Geometry of integrable models in: Fundamental’naya i Prikladnaya Matematika / Journal of Mathematical Sciences (2004).

The equation dF = 0 means that the form F is closed and is equivalent to the first Maxwell equation. If the second Maxwell equation and the continuity equation for the tensor F_{ij} hold, then the Maxwell space is associated with an electromagnetic field.

Let G_{g} be the Poincaré group, i. e., the set of the Minkowski space motions (or, equivalently, the group of diffeomorphisms of manifold M that preserve the structure (M, g)). Further, let G_{F} be the group of symplectomorphisms of the structure (M, F). By G_{S} we denote the group of diffeomorphisms of the manifold M that preserve both g and F: G_{S} = G_{g} ∩ G_{F}. Note that G_{S} is a subgroup of G_{g}. The Maxwell spaces with non-trivial groups G_{S} are interesting, for example, in connection with the well-known method for obtaining the first integrals of the Lorentz equations ([2]).

The electromagnetic fields that admit the group G_{S} were intensively studied in 1960– 70s (see [3, 4, 5, 6, 7, 8]). In [3, 4, 5], maximal subgroups of the Poincaré group that preserve the tensor F_{ij} (relativistic symmetry groups) were found for particular fields F_{ij} (homogeneous fields, plane waves, etc.) and the structure of these subgroups was studied. In [6, 7, 8], connected subgroups of the Poincaré group that are invariant transformation groups of electromagnetic fields (or, equivalently, relativistic symmetry groups) were studied. In particular, it was proved that the dimension of such a group is not greater than 6, see [8], and the classification of these groups was obtained ([6, 7]). The problem of classification with respect to conjugation for connected subgroups of the Poincaré group up to the was solved in [9] without any reference to electrodynamics.

In [10, 11], the author formulated the problem of classification with respect to the groups G_{S} for the Einstein–Maxwell spaces and related this problem with the method of obtaining the first integrals of the Lorentz equations ([2]). An Einstein–Maxwell space is a more generic object than a Maxwell space: in this case, g is a pseudo-Riemannian metric such that the pair (M, g) is an Einstein space ([12]). In case the metric be flat, G_{g} is the Poincaré group and G_{S} is its subgroup. Hence we obtain the classification problem for the Maxwell spaces with respect subgroups of the Poincaré group.

1. Formulation of the problem and method of its solution

First, we note that the class of the Maxwell spaces admitting a group G_{S} can be described as follows. Let L_{S} be the Lie algebra of vector fields corresponding to the group G_{S}. The tensor F_{ij} that defines this class is a solution of the first Maxwell equation

∂_{i}F_{jk} + ∂_{j}F_{ki} + ∂_{k}F_{ij} = 0 (i, j, k = 1, …, 4)  (1)

and is a solution of the invariance conditions with respect to G_{S} for F_{ij}

L_{ξ_{α}}F_{ij} = 0 (α = 1, …, p = dimL_{S}),  (2)

where ξ_{α} are basis vectors in L_{S} and L_{ξ_{α}} is the Lie derivative.

Note that since equations (1) and (2) are linear, the set of solutions of this system is a linear space (class) each of whose elements F_{ij} defines a Maxwell space with a symmetry group that is not less than G_{S}. But in reality for some of the Maxwell spaces of this class (sometimes, for all of them) their symmetry groups are wider than the group G_{S}. Therefore, to find the genuine symmetry group we must solve the equation L_{ξ}F_{ij} = 0 with respect to ξ ∈ L_{g}^{1} for a given F_{ij}. The space of its solutions is the Lie algebra of the symmetry group.

Further, we note that the Poincaré group has infinitely many subgroups; therefore, it is impossible to list all classes of the Maxwell spaces that are invariant under G_{S}. However, we can consider a finite number of subgroups, typical in some sense, and describe classes of Maxwell spaces for them. The list of representatives of classes of conjugate subgroups of the Poincaré group is a naturally obtained set. Indeed, if subgroup G′_{S} is conjugate to G_{S}, then there is a transformation of the coordinates A ∈ G_{g}, x^{i} = A_{i′}^{i}x^{i′} + a^{i} such that G′_{S} = A^{−1}G_{S}A. This means that the class of the Maxwell spaces with the symmetry group G_{S} defined by the tensor F_{ij} transforms into the class with the symmetry group G′_{S} defined by the tensor F′_{i′j′} = F_{i′j′} = F_{ij}A_{i′}^{i}A_{j′}^{j}.

We use the classification of subgroups of the Poincaré group up to conjugation presented in [9]. The list of the subalgebras of dimensions 1–6 of the Lie algebra L_{g} of the Poincaré group contains 76 items, some of which contain more then one algebra^{2}. We denote this algebras by L_{p,q}, where p is the dimension of the algebra and q is its number in the list of subalgebras of dimension p. If necessary, we add the symbol a, b, c, … to the notation. The problem of description of classes of the Maxwell spaces reduces to solving equations (1)–(2) for each subalgebra L_{p,q}.

Denote by C_{p,q} the class of the Maxwell spaces that corresponds to an algebra L_{p,q}. Obviously, if algebras L_{p_{1},q_{1}} and L_{p_{2},q_{2}} are such that L_{p_{1},q_{1}} ⊂ L_{p_{2},q_{2}}, (p_{1} < p_{2}), then the corresponding classes are such that C_{p_{2},q_{2}} ⊂ C_{p_{1},q_{1}}. Take the basis of the Lie algebra L_{g} in the form

e_{1} = (1, 0, 0, 0), e_{2} = (0, 1, 0, 0), e_{3} = (0, 0, 1, 0), e_{4} = (0, 0, 0, 1),
e_{12} = (−x^{2}, x^{1}, 0, 0), e_{13} = (x^{3}, 0, −x^{1}, 0), e_{23} = (0, −x^{3}, x^{2}, 0),
e_{14} = (x^{4}, 0, 0, x^{1}), e_{24} = (0, x^{4}, 0, x^{2}), e_{34} = (0, 0, x^{4}, x^{3}),

where {x^{i}} are the Galilean coordinates such that g_{ij} = diag(−1, −1, −1, 1). In what follows, L{ξ_{1}, …, ξ_{p}} is the linear combination of vectors ξ_{1}, …, ξ_{p}.

In [13, 14, 15, 16, 17, 18, 19, 20] classes of the Maxwell spaces were described in some particular cases. In [13], 22 classes of static Maxwell’s spaces^{3} were described (the smallest among those subalgebras is L_{S}^{min} = L{e_{4}}). In [14, 15], 9 classes of the Maxwell spaces admitting hyperbolic helices were described (L_{S}^{min} = L{e_{24} +λe_{1} +μe_{3}}, λ, μ = const). In [16], 15 classes of the Maxwell spaces admitting elliptic helices were described (L_{S}^{min} = L{e_{13} + λe_{2} + μe_{4}}). In [17], 18 classes of the Maxwell spaces admitting parabolic rotation were described (L_{S}^{min} = L{e_{12} − e_{14}}). In [18], 9 classes of the Maxwell spaces admitting translations along isotropic straight line were described (L_{S}^{min} = L{e_{2} + e_{4}}). In [19], 10 classes of the Maxwell spaces admitting proportional bi-rotations were described (L_{S}^{min} = L{e_{13} + λe_{24}}). In [20], 10 classes of the Maxwell spaces admitting translations were described.

1L_{g} is the Lie algebra of vector fields corresponding to the group G_{g}; it consists of the vectors ξ^{i} = a_{·j}^{i}x^{j} + b^{i}, where a_{·j}^{i} = g^{ik}a_{kj} and a_{kj} = −a_{jk}, b^{i} are real numbers.

2In fact, the number of these algebras is infinite; for example, one-dimensional subgroups of elliptic helices, parabolic helices, hyperbolic helices, and proportional bi-rotations depend on real parameters, and these subgroups are not conjugate within pairs.

3A static Maxwell’s space admits the group of translations along the time axis.

In the monograph [21], classes of the Maxwell spaces admitting subgroups of the Poincaré group were described for the whole list of subgroups (subalgebras) in [9]; this classification includes all results mentioned in the previous section. However, in all mentioned publications, there are no examples of the Maxwell spaces admitting the group G_{p,q} exactly; we obtain such examples in this article.

If the evident description of the class C_{p,q} of Maxwell spaces is absent, then we proceed as follows. For the group G_{p,q} (the algebra L_{p,q}) we describe the class P_{p,q} of potentials (covector fields A_{i} such that F_{ij} = ∂_{i}A_{j} − ∂_{j}A_{i}) is invariant with respect to this group, see [22, 23]; each A_{i} ∈ P_{p,q} must satisfy the equations L_{ξ_{α}}A_{i} = 0 (α = 1, …, p). Further, we find the subset in P_{p,q} such that each of its elements admits the group G_{p,q} exactly. Then we get the tensor F_{ij} corresponding to the potential A_{i} and verify the equality L_{S} = L_{p,q}; usually, as a result, we receive conditions for the tensor F_{ij} when the corresponding Maxwell space admits the group G_{p,q} exactly.

Remark 1. We denote groups, algebras, and classes of the Maxwell spaces as in [21]. In addition, suppose that components of the tensor F_{ij} always correspond to the Galilean coordinates {x^{i}} even if they are expressed as functions of other coordinates.

In the transversal case the Maxwell space admits the trivial symmetry group G_{S} = {id}. We find the example of the Maxwell space with the trivial group G_{S}.

Example 1. We take the potential A_{i} = (0, 0, 0, Φ), where Φ = Φ(x^{1}, x^{2}, x^{3}, x^{4}). Let Φ_{i} = ∂_{i}Φ and Φ_{ij} = ∂_{i}∂_{j}Φ. Solving equation L_{ξ}A_{i} = 0 for ξ ∈ L_{g}, we deduce that the group G_{A} ∩ G_{g} is trivial whenever functions Φ_{i}, x^{2}Φ_{1} − x^{1}Φ_{2}, x^{3}Φ_{1} − x^{1}Φ_{3}, and x^{3}Φ_{2} − x^{2}Φ_{3} are linearly independent. (For example, this condition is satisfied for the function Φ = x^{1}x^{2}x^{3}x^{4}.) Finding the algebra L_{S} for the corresponding field F_{ij}

F_{12} = F_{13} = F_{23} = 0, F_{14} = Φ_{1}, F_{24} = Φ_{2}, F_{34} = Φ_{3},  (3)

we obtain that the sufficient condition for the group G_{S} of the Maxwell space defined by (3) to be trivial is that functions Φ_{1}, Φ_{2}, Φ_{12}, Φ_{13}, Φ_{14}, x^{2}Φ_{11} − x^{1}Φ_{12}, x^{3}Φ_{11} − x^{1}Φ_{13}, x^{3}Φ_{12} − x^{2}Φ_{13} be linearly independent. In particular, this condition is provided for the function Φ = x^{1}x^{2}x^{3}x^{4}.

2. The classes of the Maxwell spaces that admit one-dimensional symmetry groups

In this section, we describe classes of the Maxwell spaces admitting the one-dimensi-onal groups from the list in [9]. Representatives of these classes are presented.

2.1. Translations. There are three types of non-conjugate in pairs, one-dimentional subgroups of translations.

2.1.1. The class C_{1,1a}. The algebra L_{1,1a} = L{e_{1}} corresponds to the one-dimensional group G_{1,1a} of translations along the space-like vector e_{1}. The class C_{1,1a} of the Maxwell spaces is defined by the tensor F_{ij} whose components satisfy the following system:

\begin{matrix}∂_{2}F_{31} + ∂_{3}F_{12} = 0, ∂_{2}F_{41} + ∂_{4}F_{12} = 0, ∂_{3}F_{41} + ∂_{4}F_{13} = 0, \\ ∂_{2}F_{34} + ∂_{3}F_{42} + ∂_{4}F_{23} = 0 (F_{ij} = F_{ij}(x^{2}, x^{3}, x^{4})) . \end{matrix}  (4)

Let us note that the class P_{1,1a} of potentials admitting the subgroup G_{1,1a} consists of the following covector fields, A_{i} = A_{i}(x^{2}, x^{3}, x^{4}). For A_{i} = (0, 0, 0, Φ), where Φ = Φ(x^{2}, x^{3}, x^{4}), we get the following tensor F_{ij}:

F_{12} = F_{13} = F_{23} = F_{14} = 0, F_{24} = ∂_{2}Φ, F_{34} = ∂_{3}Φ.  (5)

Proposition 1. Let Φ_{i} = ∂_{i}Φ and Φ_{ij} = ∂_{i}∂_{j}Φ. If the partial derivatives Φ_{2}, Φ_{3} are linearly independent and the functions Φ_{23}, Φ_{24}, x^{3}Φ_{22} − x^{2}Φ_{23} − Φ_{3} are also linearly independent, then the Maxwell space defined by the tensor (5) admits the one-dimensi-onal group G_{S} = G_{1,1a}.

In particular, these conditions are satisfied for the function Φ = x^{2}x^{3}x^{4}.

2.1.2. The class C_{1,1b}. The algebra L_{1,1b} = L{e_{4}} corresponds to the one-dimensional group G_{1,1b} of translations along the time-like vector e_{4}. The class C_{1,1b} of the Maxwell spaces is defined by the tensor F_{ij} whose components satisfy the following system:

\begin{matrix}∂_{1}F_{24} − ∂_{2}F_{14} = 0, ∂_{1}F_{34} − ∂_{3}F_{14} = 0, ∂_{2}F_{34} − ∂_{3}F_{24} = 0, \\ ∂_{1}F_{23} + ∂_{2}F_{31} + ∂_{3}F_{12} = 0 (F_{ij} = F_{ij}(x^{1}, x^{2}, x^{3})). \end{matrix}  (6)

The Maxwell spaces of the class C_{1,1b} are called static spaces. In classical vector notations

E = (E_{1}, E_{2}, E_{3}) = (F_{41}, F_{42}, F_{43}), H = (H_{1}, H_{2}, H_{3}) = (F_{32}, F_{13}, F_{21}),  (7)

the system (6) has the form rot E = 0, div H = 0. Therefore, in particular, the class C_{1,1b} includes all electrostatic and magnetostatic fields.

Let us note that the class P_{1,1b} of potentials admitting the subgroup G_{1,1b} consists of the covector fields A_{i} = A_{i}(x^{1}, x^{2}, x^{3}). For A_{i} = (0, 0, 0, Φ), Φ = Φ(x^{1}, x^{2}, x^{3}), we get the following tensor F_{ij} of electrostatic field:

F_{12} = F_{13} = F_{23} = 0, F_{α4} = ∂_{α}Φ (α = 1, 2, 3).  (8)

Proposition 2. Let Φ_{i} = ∂_{i}Φ and Φ_{ij} = ∂_{i}∂_{j}Φ. If derivatives Φ_{1}, Φ_{2} are linearly independent and functions Φ_{12}, Φ_{13}, x^{2}Φ_{11} − x^{1}Φ_{12} − Φ_{2}, x^{3}Φ_{11} − x^{1}Φ_{13} − Φ_{3}, x^{3}Φ_{12} − x^{2}Φ_{13} are also linearly independent, then the Maxwell space defined by the tensor (8) admits the one-dimensional group G_{S} = G_{1,1b}.

In particular, these conditions are satisfied for the function Φ = x^{1}x^{2}x^{3}.

2.1.3. The class C_{1,1c}. The algebra L_{1,1c} = L{e_{2} + e_{4}} corresponds to the one-dimensi-onal group G_{1,1c} of translations along the isotropic vector e_{2}+e_{4}. We use the substitution

v^{1} = x^{1}, v^{2} = x^{2} + x^{4}, v^{3} = x^{3}, v^{4} = x^{2} − x^{4}.  (9)

The class C_{1,1c} of the Maxwell spaces is defined by the tensor F_{ij} = F_{ij}(v^{1}, v^{3}, v^{4}) whose components satisfy the following system:

\begin{matrix}\frac{∂F_{23}}{∂v^{1}} + \frac{∂F_{12}}{∂v^{3}} − \frac{∂F_{13}}{∂v^{4}} = 0, \frac{∂F_{24}}{∂v^{1}} − \frac{∂F_{12}}{∂v^{4}} − \frac{∂F_{14}}{∂v^{4}} = 0, \\ \frac{∂F_{34}}{∂v^{1}} − \frac{∂F_{14}}{∂v^{3}} − \frac{∂F_{13}}{∂v^{4}} = 0, \frac{∂F_{34}}{∂v^{4}} − \frac{∂F_{23}}{∂v^{4}} − \frac{∂F_{24}}{∂v^{3}} = 0. \end{matrix}  (10)

Note that the class P_{1,1c} of potentials admitting the subgroup G_{1,1c} consists of the covector fields A_{i} = A_{i}(v^{1}, v^{3}, v^{4}). For the potential A_{i} = (0, 0, 0, Φ), where Φ = Φ(v^{1}, v^{3}, v^{4}) = Φ(x^{1}, x^{3}, x^{2} − x^{4}), we obtain

F_{12} = F_{13} = F_{23} = 0, F_{14} = \frac{∂Φ}{∂v^{1}} , F_{24} = \frac{∂Φ}{∂v^{4}} , F_{34} = \frac{∂Φ}{∂v^{3}} .  (11)

Proposition 3. Let Φ_{i} = ∂Φ/∂v^{i} and Φ_{ij} = ∂^{2}Φ/∂v^{i}∂v^{j}. If partial derivatives Φ_{1}, Φ_{4} are linearly independent and the functions Φ_{13}, Φ_{14}, x^{2}Φ_{11} − x^{1}Φ_{14} − Φ_{4}, x^{3}Φ_{11} − x^{1}Φ_{13} −Φ_{3}, x^{3}Φ_{14} −x^{2}Φ_{13} are also linearly independent, then the Maxwell space defined by the tensor (11) admits the one-dimensional group G_{S} = G_{1,1c}.

For example, it is so for the function Φ = v^{1}v^{3}v^{4} = x^{1}x^{3}(x^{2} − x^{4}).

2.2. Elliptic helices. The algebra L_{1,2} = L{e_{13} + λe_{2} + μe_{4}} corresponds to the onedimensional group G_{1,2} of elliptic helices as follows

x̂^{1} = x^{1} cos a + x^{3} sin a, x̂^{2} = λa + x^{2},

x̂^{3} = −x^{1} sin a + x^{3} cos a, x̂^{4} = μa + x^{4}. (el)

If λ = μ = 0, then transformations (el) are rotations. If λ ≠ 0, μ = 0, then transformations (el) are helices with the space-like axis. If λ = 0, μ ≠ 0, then transformations (el) are helices with the time-like axis. If λ = μ ≠ 0, then transformations (el) are helices with the isotropic axis. We use the coordinate system {x̃^{i}} = {r, x̃^{2}, φ, x̃^{4}}, connected with {x^{i}} by the formulas

x^{1} = r sin φ, x^{2} = λφ + x̃^{2}, x^{3} = r cos φ, x^{4} = μφ + x̃^{4}.  (12)

The class C_{1,2} of the Maxwell spaces is defined by the tensor

F_{12} = c_{1} cos φ + c_{2} sin φ, F_{13} = F_{13}(r, x̃^{2}, x̃^{4}),
F_{14} = c_{3} cos φ + c_{4} sin φ, F_{23} = c_{1} sin φ − c_{2} cos φ,
F_{24} = F_{24}(r, x̃^{2}, x̃^{4}),  F_{34} = −c_{3} sin φ + c_{4} cos φ,

(13)

where c_{i} = c_{i}(r, x̃^{2}, x̃^{4}) are smooth functions which satisfy the following system of equations

\begin{matrix}\frac{∂c_{1}}{∂r} + \frac{c_{1}}{r} + \frac{λ}{r} \frac{∂c_{2}}{∂x̃^{2}} + \frac{μ}{r} \frac{∂c_{2}}{∂x̃^{4}} − \frac{∂F_{13}}{∂x̃^{2}} = 0, \frac{∂F_{24}}{∂r} + \frac{∂c_{2}}{∂x̃^{4}} − \frac{∂c_{4}}{∂x̃^{2}} = 0, \\ \frac{∂c_{3}}{∂r} + \frac{c_{3}}{r} + \frac{λ}{r} \frac{∂c_{4}}{∂x̃^{2}} + \frac{μ}{r} \frac{∂c_{4}}{∂x̃^{4}} − \frac{∂F_{13}}{∂x̃^{4}} = 0, \\ \frac{λ}{r} \frac{∂F_{24}}{∂x̃^{2}} + \frac{μ}{r} \frac{∂F_{24}}{∂x̃^{4}} + \frac{∂c_{3}}{∂x̃^{2}} − \frac{∂c_{1}}{∂x̃^{4}} = 0. \end{matrix}  (14)

It is easily proved that the class P_{1,2} of potentials A_{i} that is invariant with respect to the group G_{1,2} consists of the fields

A_{1} = C_{1} cos φ + C_{2} sin φ, A_{2} = A_{2}(r, x̃^{2}, x̃^{4}),

A_{3} = −C_{1} sin φ + C_{2} cos φ, A_{4} = A_{4}(r, x̃^{2}, x̃^{4}), (15)

where C_{i} = C_{i}(r, x̃^{2}, x̃^{4}). For the potential A_{i} = (0, 0, 0, Φ), Φ = Φ(u^{1}, u^{2}, u^{3}) = Φ(r, x̃^{2}, x̃^{4}), we obtain

F_{12} = F_{13} = 0, F_{23} = 0, F_{24} = Φ_{2},

F_{14} = Φ_{1} sin φ − \frac{λ}{r}Φ_{2} cos φ − \frac{μ}{r}Φ_{3} cos φ,

F_{34} = Φ_{1} cos φ + \frac{λ}{r}Φ_{2} sin φ + \frac{μ}{r}Φ_{3} sin φ,

(16)

where Φ_{k} = ∂Φ/∂u^{k}. Consider four cases, which are different from the geometric point of view.

2.2.1. The class C_{1,2a} (μ = 0, λ ≠ 0). The algebra L_{1,2a} = L{e_{13} + λe_{2}} corresponds to the group G_{1,2a} of elliptic helices with the space-like axis Ox^{2}. Equations (14) and the tensor (16) obtain the form

\begin{matrix}\frac{∂c_{1}}{∂r} + \frac{c_{1}}{r} + \frac{λ}{r} \frac{∂c_{2}}{∂x̃^{2}} − \frac{∂F_{13}}{∂x̃^{2}} = 0, \frac{∂F_{24}}{∂r} + \frac{∂c_{2}}{∂x̃^{4}} − \frac{∂c_{4}}{∂x̃^{2}} = 0, \\ \frac{∂c_{3}}{∂r} + \frac{c_{3}}{r} + \frac{λ}{r} \frac{∂c_{4}}{∂x̃^{2}} − \frac{∂F_{13}}{∂x̃^{4}} = 0, \frac{λ}{r} \frac{∂F_{24}}{∂x̃^{2}} + \frac{∂c_{3}}{∂x̃^{2}} − \frac{∂c_{1}}{∂x̃^{4}} = 0 \end{matrix}  (17)

and

F_{12} = F_{13} = 0, F_{14} = Φ_{1} sin φ − \frac{λ}{r}Φ_{2} cos φ,

F_{23} = 0, F_{24} = Φ_{2}, F_{34} = Φ_{1} cos φ + \frac{λ}{r}Φ_{2} sin φ.

(18)

Proposition 4. Let Φ_{i} = ∂Φ/∂v^{i} and Φ_{ij} = ∂^{2}Φ/∂v^{i}∂v^{j}. If partial derivatives Φ_{22}, Φ_{23} are linearly independent and Φ_{21} ≠ 0, then the Maxwell space defined by the tensor (18) admits the one-dimensional group G_{S} = G_{1,2a}.

2.2.2. The class C_{1,2b} (λ = 0, μ ≠ 0). The algebra L_{1,2b} = L{e_{13} + μe_{4}} corresponds to the group G_{1,2b} of elliptic helices with the time-like axis Ox^{4}. Equations (14) and the tensor (16) obtain the form

\begin{matrix}\frac{∂c_{1}}{∂r} + \frac{c_{1}}{r} + \frac{μ}{r} \frac{∂c_{2}}{∂x̃^{4}} − \frac{∂F_{13}}{∂x̃^{2}} = 0, \frac{∂F_{24}}{∂r} + \frac{∂c_{2}}{∂x̃^{4}} − \frac{∂c_{4}}{∂x̃^{2}} = 0, \\ \frac{∂c_{3}}{∂r} + \frac{c_{3}}{r} + \frac{μ}{r} \frac{∂c_{4}}{∂x̃^{4}} − \frac{∂F_{13}}{∂x̃^{4}} = 0, \frac{μ}{r} \frac{∂F_{24}}{∂x̃^{4}} + \frac{∂c_{3}}{∂x̃^{2}} − \frac{∂c_{1}}{∂x̃^{4}} = 0 \end{matrix}  (19)

and

F_{12} = F_{13} = 0, F_{14} = Φ_{1} sin φ − \frac{μ}{r}Φ_{3} cos φ,

F_{23} = 0, F_{24} = Φ_{2}, F_{34} = Φ_{1} cos φ + \frac{μ}{r}Φ_{3} sin φ.

(20)

Proposition 5. Let Φ_{i} = ∂Φ/∂v^{i} and Φ_{ij} = ∂^{2}Φ/∂v^{i}∂v^{j}. If the partial derivatives Φ_{22}, Φ_{23} are linearly independent and Φ_{21} ≠ 0, then the Maxwell space defined by the tensor (20) admits the one-dimensional group G_{S} = G_{1,2b}.

2.2.3. The class C_{1,2c} (λ = μ ≠ 0). The algebra L_{1,2c} = L{e_{13} +λ(e_{2} +e_{4})} corresponds to the group G_{1,2c} of elliptic helices with the isotropic axis. Equations (14) and the tensor (16) acquire the form

\begin{matrix}\frac{∂c_{1}}{∂r} + \frac{c_{1}}{r} + \frac{λ}{r} (\frac{∂c_{2}}{∂x̃^{2}} + \frac{∂c_{2}}{∂x̃^{4}} ) − \frac{∂F_{13}}{∂x̃^{2}} = 0, \frac{∂F_{24}}{∂r} + \frac{∂c_{2}}{∂x̃^{4}} − \frac{∂c_{4}}{∂x̃^{2}} = 0, \\ \frac{∂c_{3}}{∂r} + \frac{c_{3}}{r} + \frac{λ}{r} (\frac{∂c_{4}}{∂x̃^{2}} + \frac{∂c_{4}}{∂x̃^{4}} ) − \frac{∂F_{13}}{∂x̃^{4}} = 0, \\ \frac{λ}{r} (\frac{∂F_{24}}{∂x̃^{2}} + \frac{∂F_{24}}{∂x̃^{4}} ) + \frac{∂c_{3}}{∂x̃^{2}} − \frac{∂c_{1}}{∂x̃^{4}} = 0 \end{matrix}  (21)

and

F_{12} = F_{13} = 0, F_{14} = Φ_{1} sin φ − \frac{λ}{r} cos φ(Φ_{2} + Φ_{3}),

F_{23} = 0, F_{24} = Φ_{2}, F_{34} = Φ_{1} cos φ + \frac{λ}{r} sin φ(Φ_{2} + Φ_{3}).

(22)

Proposition 6. Let Φ_{i} = ∂Φ/∂v^{i} and Φ_{ij} = ∂^{2}Φ/∂v^{i}∂v^{j}. If Φ_{21} ≠ 0 and Φ_{22}+Φ_{23} ≠ 0, then the Maxwell space defined by the tensor (22) admits the one-dimensional group G_{S} = G_{1,2c}.

2.2.4. The class C_{1,2d} (λ = μ = 0). The algebra L_{1,2d} = L{e_{13}} corresponds to the group G_{1,2d} of rotations in the plane Ox^{1}x^{3}. Equations (14) and the tensor (16) have the form

\begin{matrix}\frac{∂c_{1}}{∂r} + \frac{c_{1}}{r} − \frac{∂F_{13}}{∂x̃^{2}} = 0, \frac{∂F_{24}}{∂r} + \frac{∂c_{2}}{∂x̃^{4}} − \frac{∂c_{4}}{∂x̃^{2}} = 0, \\ \frac{∂c_{3}}{∂r} + \frac{c_{3}}{r} − \frac{∂F_{13}}{∂x̃^{4}} = 0, \frac{∂c_{3}}{∂x̃^{2}} − \frac{∂c_{1}}{∂x̃^{4}} = 0 \end{matrix}  (23)

and

F_{12} = F_{13} = F_{23} = 0, F_{14} = Φ_{1} sin φ, F_{24} = Φ_{2}, F_{34} = Φ_{1} cos φ.  (24)

Proposition 7. Let Φ_{i} = ∂Φ/∂v^{i} and Φ_{ij} = ∂^{2}Φ/∂v^{i}∂v^{j}. If x̃^{2}Φ_{21} − rΦ_{22} + Φ_{1} ≠ 0, Φ_{21} ≠ 0, and partial derivatives Φ_{22}, Φ_{23} are linearly independent, then the Maxwell space defined by the tensor (24) admits the one-dimensional group G_{S} = G_{1,2d}.

2.3. Hyperbolic helices. The algebra L_{1,3} = L{e_{24} + λe_{1}} corresponds to the group G_{1,3} of hyperbolic helices with the axis Ox^{1}:

x̂^{1} = x^{1} + λa, x̂^{2} = x^{2} cosh a + x^{4} sinh a,

x̂^{3} = x^{3}, x̂^{4} = x^{2} sinh a + x^{4} cosh a. (hyp)

If λ = 0, then (hyp) are the Lorentz transformations. We use the coordinate system {x̃^{i}} = {x̃^{1}, r, x̃^{3}, φ}, related to {x^{i}} by the formulae

x^{1} = λφ + x̃^{1}, x^{2} = r cosh φ, x^{3} = x̃^{3}, x^{4} = r sinh φ.  (25)

2.3.1. The class C_{1,3} (λ ≠ 0) of the Maxwell spaces is defined by the following tensor F_{ij}

F_{12} = c_{1} cosh φ + c_{2} sinh φ, F_{13} = F_{13}(x̃^{1}, r, x̃^{3}),
F_{14} = −c_{1} sinh φ − c_{2} cosh φ, F_{23} = c_{3} cosh φ + c_{4} sinh φ,
F_{24} = F_{24}(x̃^{1}, r, x̃^{3}), F_{34} = c_{3} sinh φ + c_{4} cosh φ,

(26)

where c_{i} = c_{i}(x̃^{1}, r, x̃^{3}) (i = 1, …, 4) are smooth functions which satisfy the system

\begin{matrix}\frac{∂c_{2}}{∂r} + \frac{c_{2}}{r} − \frac{λ}{r} \frac{∂c_{1}}{∂x̃^{1}} + \frac{∂F_{24}}{∂x̃^{1}} = 0, \frac{∂c_{1}}{∂x̃^{3}} − \frac{∂F_{13}}{∂r} + \frac{∂c_{3}}{∂x̃^{1}} = 0, \\ \frac{∂c_{4}}{∂r} + \frac{c_{4}}{r} − \frac{λ}{r} \frac{∂c_{3}}{∂x̃^{1}} − \frac{∂F_{24}}{∂x̃^{3}} = 0, \frac{λ}{r} \frac{∂F_{13}}{∂x̃^{1}} − \frac{∂c_{4}}{∂x̃^{1}} − \frac{∂c_{2}}{∂x̃^{3}} = 0. \end{matrix}  (27)

Note that the class P_{1,3} of potentials A_{i} that is invariant with respect to the group G_{1,3} consists of the fields [23]

A_{1} = A_{1}(x̃^{1}, r, x̃^{3}), A_{2} = C_{1} cosh φ + C_{2} sinh φ,

A_{3} = A_{3}(x̃^{1}, r, x̃^{3}), A_{4} = −C_{1} sinh φ − C_{2} cosh φ, (28)

where C_{i} = C_{i}(x̃^{1}, r, x̃^{3}). For the potential A_{i} = (−Φ, 0, 0, 0), Φ = Φ(x̃^{1}, x̃^{3}), we obtain

F_{12} = \frac{λ}{r}Φ_{1} sinh φ, F_{13} = Φ_{3}, F_{14} = −\frac{λ}{r}Φ_{1} cosh φ, F_{23} = F_{24} = F_{34} = 0, (29)

where Φ_{i} = ∂Φ/∂x̃^{i} (i = 1, 3). Let Φ_{ij} = ∂^{2}Φ/∂x̃^{i}∂x̃^{j}.

Proposition 8. If the partial derivatives Φ_{31}, Φ_{33} are linearly independent, then the Maxwell space defined by the tensor (29) admits the one-dimensional group G_{S} = G_{1,3}.

2.3.2. The class C_{1,3b}. For λ = 0 the class of the Maxwell spaces that corresponds to the algebra L_{1,3b} = L{e_{24}}^{3}, is defined by the tensor (26), where the functions c_{i} = c_{i}(x̃^{1}, r, x̃^{3}) must satisfy the equations (instead of (27))

\begin{matrix}\frac{∂c_{2}}{∂r} + \frac{c_{2}}{r} + \frac{∂F_{24}}{∂x̃^{1}} = 0, \frac{∂c_{1}}{∂x̃^{3}} − \frac{∂F_{13}}{∂r} + \frac{∂c_{3}}{∂x̃^{1}} = 0, \\ \frac{∂c_{4}}{∂r} + \frac{c_{4}}{r} − \frac{∂F_{24}}{∂x̃^{3}} = 0, \frac{∂c_{4}}{∂x̃^{1}} + \frac{∂c_{2}}{∂x̃^{3}} = 0; \end{matrix}  (30)

we use the Lorentz transformation

x^{1} = x̃^{1}, x^{2} = r cosh φ, x^{3} = x̃^{3}, x^{4} = r sinh φ.  (31)

instead of (25).

For the potential A_{i} = (Φ, 0, 0, 0), Φ = Φ(x̃^{1}, r, x̃^{3}), we have

F_{12} = Φ_{r} cosh φ, F_{13} = Φ_{3}, F_{14} = −Φ_{r} sinh φ, F_{23} = F_{24} = F_{34} = 0 (32)

(Φ_{r} = ∂Φ/∂r). Let Φ_{ij} = ∂^{2}Φ/∂x̃^{i}∂x̃^{j}.

3We do not present here the class C_{1,3a} that corresponds to the algebra L_{1,3a} = L{e_{24} + λe_{3}} since the groups G_{1,3a} and G_{1,3} are conjugated; in [14, 15, 21], this class is used for description of other classes.

Proposition 9. If the partial derivatives Φ_{31}, Φ_{3r}, Φ_{33} are linearly independent, then the Maxwell space defined by the tensor (32) admits the one-dimensional group G_{S} = G_{1,3}.

2.4. Parabolic helices. Here we describe the class C_{1,4} of the Maxwell spaces that corresponds to the algebra L_{1,4} = L{e_{12} − e_{14} + λe_{2} + μe_{3}} (λ, μ = const, λμ = 0). This class is defined by the pair of equations (1) and (2) for the vector ξ = e_{12}−e_{14}+λe_{2}+μe_{3}; the second equation is the system

\begin{matrix}XF_{12} + F_{24} = 0, XF_{13} + F_{23} + F_{34} = 0, XF_{14} + F_{24} = 0, \\ XF_{23} − F_{13} = 0, XF_{24} + F_{12} − F_{14} = 0, XF_{34} + F_{13} = 0, \end{matrix}  (33)

where

X = −(x^{2} + x^{4})∂_{1} + (x^{1} + λ)∂_{2} + μ∂_{3} − x^{1}∂_{4}.  (34)

We consider three cases: a) λ = μ = 0; b) λ = 0, μ ≠ 0; c) λ ≠ 0, μ = 0.

2.4.1. The class C_{1,4a}. The group G_{1,4a} corresponding to the algebra L_{1,4a} = L{e_{12}−e_{14}} consists of parabolic rotations. We have

x̂^{1} = x^{1} − a(x^{2} + x^{4}),  x̂^{2} = x^{2} + ax^{1} − \frac{a^{2}}{2} (x^{2} + x^{4}),
x̂^{3} = x^{3},  x̂^{4} = x^{4} − ax^{1} + \frac{a^{2}}{2} (x^{2} + x^{4}).

Let {x̃^{i}} be the coordinate system related with {x^{i}} by formulae

x̃^{1} = x^{2} + x^{4}, x̃^{2} = −\frac{x^{1}}{x^{2} + x^{4}} , x̃^{3} = x^{3}, x̃^{4} = \frac{1}{2}(x^{1})^{2} + x^{2}(x^{2} + x^{4}). (35)

Then the operator (34) is replaced by partial derivative with respect to x̃^{2} and the solution of the system that follows from (33) is defined by the formulae

\begin{matrix}F_{13} = C_{1}x̃^{2} + C_{2},  F_{24} = C_{5}x̃^{2} + C_{6}, \\ F_{23} = \frac{C_{1}}{2} (x̃^{2})^{2} + C_{2}x̃^{2} + C_{3}, F_{12} = −\frac{C_{5}}{2} (x̃^{2})^{2} − C_{6}x̃^{2} + C_{7}, \\ F_{34} = −\frac{C_{1}}{2} (x̃^{2})^{2} − C_{2}x̃^{2} + C_{4}, F_{14} = −\frac{C_{5}}{2} (x̃^{2})^{2} − C_{6}x̃^{2} + C_{8}, \end{matrix}  (36)

where C_{k} = C_{k}(x̃^{1}, x̃^{3}, x̃^{4}) are smooth functions that satisfy the equations

C_{1} + C_{3} + C_{4} = 0, C_{5} + C_{7} − C_{8} = 0.  (37)

Substituting (36) in the Maxwell equations (1), taking into account the independence of functions C_{k} from x̃^{2}, grouping the summands with equal powers of the variable x̃^{2}, and using the linear independence of the functions 1, x̃^{2}, (x̃^{2})^{2}, (x̃^{2})^{3}, (x̃^{2})^{4}, we finally obtain that the functions C_{k} satisfy the equations

x̃^{1} \frac{∂C_{2}}{∂x̃^{4}} + \frac{∂C_{5}}{∂x̃^{3}} = 0,  (38a)
x̃^{1} \frac{∂C_{3}}{∂x̃^{4}} + \frac{∂C_{1}}{∂x̃^{1}} + \frac{(x̃^{1})^{2} + x̃^{4}}{x̃^{1}} \frac{∂C_{1}}{∂x̃^{4}} + \frac{∂C_{6}}{∂x̃^{3}} = 0,  (38b)
−\frac{C_{2}}{x̃^{1}} − \frac{∂C_{2}}{∂x̃^{1}} − \frac{(x̃^{1})^{2} + x̃^{4}}{x̃^{1}} \frac{∂C_{2}}{∂x̃^{4}} + \frac{∂C_{7}}{∂x̃^{3}} = 0,  (38c)
−\frac{C_{5}}{x̃^{1}} + \frac{∂C_{7}}{∂x̃^{1}} − \frac{∂C_{8}}{∂x̃^{1}} − \frac{(x̃^{1})^{2} + x̃^{4}}{x̃^{1}} \frac{∂C_{8}}{∂x̃^{4}} + \begin{matrix} & x̃^{4} \\  & x̃^{1} \end{matrix} \frac{∂C_{7}}{∂x̃^{4}} = 0,  (38d)
−x̃^{1} \frac{∂C_{4}}{∂x̃^{4}} + \frac{∂C_{6}}{∂x̃^{3}} + \frac{∂C_{1}}{∂x̃^{1}} + \begin{matrix} & x̃^{4} \\  & x̃^{1} \end{matrix} \frac{∂C_{1}}{∂x̃^{4}} = 0,  (38e)
\frac{C_{2}}{x̃^{1}} + \frac{∂C_{2}}{∂x̃^{1}} + \begin{matrix} & x̃^{4} \\  & x̃^{1} \end{matrix} \frac{∂C_{2}}{∂x̃^{4}} − \frac{∂C_{8}}{∂x̃^{3}} = 0,  (38f)
\frac{∂C_{4}}{∂x̃^{1}} + \frac{(x̃^{1})^{2} + x̃^{4}}{x̃^{1}} \frac{∂C_{4}}{∂x̃^{4}} − \frac{∂C_{6}}{∂x̃^{3}} + \frac{∂C_{3}}{∂x̃^{1}} + \begin{matrix} & x̃^{4} \\  & x̃^{1} \end{matrix} \frac{∂C_{3}}{∂x̃^{4}} = 0.  (38g)

Thus, the Maxwell space of the class C_{1,4a}, corresponding to the algebra L_{1,4a}, is defined by the tensor F_{ij} of the form (36) satisfying to (37) and (38).

Proposition 10. The Maxwell space defined by the tensor F_{ij} is of the form

F_{12} = F_{14} = F_{24} = 0, F_{13} = x^{1}Φ_{3},

F_{23} = Φ_{1} + (2x^{2} + x^{4})Φ_{3}, F_{34} = −Φ_{1} − x^{2}Φ_{3},

where

Φ(t_{1}, t_{2}, t_{3}) = Φ(x̃^{1}, x̃^{3}, x̃^{4}) = Φ (x^{2} + x^{4}, x^{3}, \frac{1}{2}(x^{1})^{2} + x^{2}(x^{2} + x^{4})) ,

Φ_{1} = ∂Φ/∂t_{1}, Φ_{3} = ∂Φ/∂t_{3}, admits the one-dimensional group G_{S} = G_{1,4a} whenever the partial derivatives ∂^{2}Φ/∂t_{3}∂t_{1}, ∂^{2}Φ/∂t_{3}∂t_{2} ∂^{2}Φ/∂t_{3}^{2} are linearly independent.

In particular, this condition is satisfied for the function Φ(t_{1}, t_{2}, t_{3}) = (t_{1} + t_{2}^{2} + t_{3}^{2})t_{3}.

2.4.2. The class C_{1,4b}. The group G_{1,4b} that corresponds to the algebra L_{1,4b} = L{e_{12} − e_{14} + μe_{3}} consists of the parabolic helices of the form

x̂^{1} = x^{1} − a(x^{2} + x^{4}), x̂^{2} = x^{2} + ax^{1} − \frac{a^{2}}{2} (x^{2} + x^{4}),

x̂^{3} = x^{3} + μa, x̂^{4} = x^{4} − ax^{1} + \frac{a^{2}}{2} (x^{2} + x^{4}),

which turn into parabolic rotations for μ = 0. In this case, the change {x^{i}} → {x̃^{i}} of the form

x̃^{1} = x^{2} + x^{4}, x̃^{2} = −\frac{x^{1}}{x^{2} + x^{4}} , x̃^{3} = x^{3} + \frac{μx^{1}}{x^{2} + x^{4}} , x̃^{4} = \frac{1}{2}(x^{1})^{2} + x^{2}(x^{2} + x^{4}), (39)

coinciding for μ = 0 with (35), replaces the operator (34) (λ = 0) by the partial derivative with respect to x̃^{2}. Repeating the same operations as for the class C_{1,4a}, we obtain the following result. The Maxwell space of the class C_{1,4b} is defined by the tensor F_{ij} of the form (36), where the change (35) is replaced by (39) and the functions C_{k} = C_{k}(x̃^{1}, x̃^{3}, x̃^{4}) satisfy the system of equations (37), (38b), (38e), (38g), and

\frac{μ}{x̃^{1}} \frac{∂C_{1}}{∂x̃^{3}} + x̃^{1} \frac{∂C_{2}}{∂x̃^{4}} + \frac{∂C_{5}}{∂x̃^{3}} = 0,  (40a)
\frac{μ}{x̃^{1}} \frac{∂C_{3}}{∂x̃^{3}} − \frac{C_{2}}{x̃^{1}} − \frac{∂C_{2}}{∂x̃^{1}} − \frac{(x̃^{1})^{2} + x̃^{4}}{x̃^{1}} \frac{∂C_{2}}{∂x̃^{4}} + \frac{∂C_{7}}{∂x̃^{3}} = 0,  (40b)
\frac{μ}{x̃^{1}} \frac{∂C_{6}}{∂x̃^{3}} − \frac{C_{5}}{x̃^{1}} + \frac{∂C_{7}}{∂x̃^{1}} − \frac{∂C_{8}}{∂x̃^{1}} − \frac{(x̃^{1})^{2} + x̃^{4}}{x̃^{1}} \frac{∂C_{8}}{∂x̃^{4}} + \begin{matrix} & x̃^{4} \\  & x̃^{1} \end{matrix} \frac{∂C_{7}}{∂x̃^{4}} = 0,  (40c)
\frac{μ}{x̃^{1}} \frac{∂C_{4}}{∂x̃^{3}} + \frac{C_{2}}{x̃^{1}} + \frac{∂C_{2}}{∂x̃^{1}} + \begin{matrix} & x̃^{4} \\  & x̃^{1} \end{matrix} \frac{∂C_{2}}{∂x̃^{4}} − \frac{∂C_{8}}{∂x̃^{3}} = 0.  (40d)

Note that for μ = 0 the last four equations coincide with (38a), (38c), (38d), and (38f), respectively.

Proposition 11. The Maxwell space defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{24} = 0, F_{13} = \frac{μ}{x^{2} + x^{4}} Φ_{2} + x^{1}Φ_{3},

F_{23} = Φ_{1} − \frac{μx^{1}}{(x^{2} + x^{4})^{2}} Φ_{2} + (2x^{2} + x^{4})Φ_{3},

F_{34} = −Φ_{1} + \frac{μx^{1}}{(x^{2} + x^{4})^{2}} Φ_{2} − x^{2}Φ_{3},

where

Φ(t_{1}, t_{2}, t_{3}) = Φ (x^{2} + x^{4}, x^{3} + \frac{μx^{1}}{x^{2} + x^{4}} Φ_{2}, \frac{1}{2}(x^{1})^{2} + x^{2}(x^{2} + x^{4})) ,

Φ_{i} = ∂Φ/∂t_{i}, admits the one-dimensional group G_{S} = G_{1,4b} whenever the partial derivatives ∂^{2}Φ/∂t_{3}∂t_{1}, ∂^{2}Φ/∂t_{3}∂t_{2} and ∂^{2}Φ/∂t_{3}^{2} are linearly independent.

In particular, this condition holds for the function Φ(t_{1}, t_{2}, t_{3}) = (t_{1} + t_{2}^{2} + t_{3}^{2})t_{3}.

2.4.3. The class C_{1,4c}. The group G_{1,4c} that corresponds to the algebra L_{1,4c} = L{e_{12} − e_{14} + λe_{2}} consists of the parabolic helices of the form

\begin{matrix}x̂^{1} = x^{1} − a(x^{2} + x^{4}) − λ\frac{a^{2}}{2} , \\ x̂^{3} = x^{3}, \end{matrix} \begin{matrix}x̂^{2} = x^{2} + a(x^{1} + λ) − \frac{a^{2}}{2} (x^{2} + x^{4}) − λ\frac{a^{3}}{6} , \\ x̂^{4} = x^{4} − ax^{1} + \frac{a^{2}}{2} (x^{2} + x^{4}) + λ\frac{a^{3}}{6} , \end{matrix}

which turn into parabolic rotations for λ = 0. The change {x^{i}} → {x̃^{i}} of the form

\begin{matrix}x̃^{1} = 2λx^{1} + (x^{2} + x^{4})^{2} , x̃^{2} = \frac{x^{2} + x^{4}}{λ} , \\ x̃^{3} = x^{3}, x̃^{4} = λx^{4} + x^{1}(x^{2} + x^{4}) + \frac{1}{3λ} (x^{2} + x^{4})^{3} , \end{matrix}  (41)

replaces the operator (34) (μ = 0) by the partial derivative with respect to x̃^{2}. Repeating the same operations as for the class C_{1,4a}, we obtain the following result. The Maxwell space of the class C_{1,4c} is defined by the tensor F_{ij} of the form (36), where the change (35) is replaced by (41) and the functions C_{k} = C_{k}(x̃^{1}, x̃^{3}, x̃^{4}) satisfy both Eq. (37) and the system

\frac{C_{1}}{λ} + \frac{x̃^{1}}{2λ} \frac{∂C_{2}}{∂x̃^{4}} − 2λ\frac{∂C_{3}}{∂x̃^{1}} − \frac{∂C_{7}}{∂x̃^{3}} = 0,  (42a)
\frac{x̃^{1}}{2λ} \frac{∂C_{1}}{∂x̃^{4}} − λ\frac{∂C_{3}}{∂x̃^{4}} + \frac{∂C_{6}}{∂x̃^{3}} = 0,  (42b)
2λ\frac{∂C_{1}}{∂x̃^{1}} − λ\frac{∂C_{2}}{∂x̃^{4}} + \frac{∂C_{5}}{∂x̃^{3}} = 0,  (42c)
\frac{x̃^{1}}{2λ} \frac{∂C_{5}}{∂x̃^{4}} − 2λ\frac{∂C_{6}}{∂x̃^{1}} − λ\frac{∂C_{7}}{∂x̃^{4}} = 0.  (42d)

Proposition 12. The Maxwell space defined by the tensor F_{ij} such that

F_{12} = F_{14} = F_{24} = 0, F_{13} = 2λΦ_{1} + (x^{2} + x^{4})Φ_{3},

F_{23} = 2(x^{2} + x^{4})Φ_{1} + (x^{1} + \frac{(x^{2} + x^{4})^{2}}{λ} ) Φ_{3},

F_{34} = −2(x^{2} + x^{4})Φ_{1} − (λ + x^{1} + \frac{(x^{2} + x^{4})^{2}}{λ} ) Φ_{3},

where

Φ(t_{1}, t_{2}, t_{3}) =

Φ (2λx^{1} + (x^{2} + x^{4})^{2} , x^{3}, λx^{4} + x^{1}(x^{2} + x^{4}) + \frac{1}{3λ} (x^{2} + x^{4})^{3}) ,

admits the one-dimensional group G_{S} = G_{1,4c} whenever the partial derivatives Φ_{i} = ∂Φ/∂t_{i}, Φ_{11}, Φ_{12}, and Φ_{13} are linearly independent (Φ_{ij} = ∂^{2}Φ/∂t_{i}∂t_{j}).

In particular, this condition holds for the function Φ(t_{1}, t_{2}, t_{3}) = t_{1}(t_{1}^{2} + t_{2}^{2} + t_{3}^{2}).

2.5. Proportional bi-rotations. The algebra L_{1,5} = L{e_{13} + λe_{24}} (λ ≠ 0) corresponds to the group G_{1,5} of proportional bi-rotations of the form

x̂^{1} = x^{3} sin a + x^{1} cos a, x̂^{2} = x^{2} cosh λa + x^{4} sinh λa,
x̂^{3} = x^{3} cos a − x^{1} sin a, x̂^{4} = x^{2} sinh λa + x^{4} cosh λa.

Here we use the coordinate system {r, ρ, θ, φ} related with {x^{i}} by the formulae

x^{1} = r cos(θ − φ), x^{2} = ρ cosh(λφ), x^{3} = r sin(θ − φ), x^{4} = ρ sinh(λφ). (43)

The class C_{1,5} of the Maxwell spaces is defined by the tensor F_{ij} of the following form

F_{12} = (−c_{1} cosh λφ − c_{2} sinh λφ) sin φ + (c_{3} cosh λφ + c_{4} sinh λφ) cos φ,
F_{14} = (c_{2} cosh λφ + c_{1} sinh λφ) sin φ − (c_{4} cosh λφ + c_{3} sinh λφ) cos φ,
F_{23} = (c_{1} cosh λφ + c_{2} sinh λφ) cos φ + (c_{3} cosh λφ + c_{4} sinh λφ) sin φ,
F_{34} = (c_{2} cosh λφ + c_{1} sinh λφ) cos φ + (c_{4} cosh λφ + c_{3} sinh λφ) sin φ,

(44)

F_{13} = F_{13}(ρ, r, θ), F_{24} = F_{24}(ρ, r, θ),  (45)

where the functions c_{i} = c_{i}(ρ, r, θ), F_{13}(ρ, r, θ), and F_{24}(ρ, r, θ) satisfy the following system:

\frac{sin θ}{r} \frac{∂F_{24}}{∂θ} − cos θ\frac{∂F_{24}}{∂r} + \frac{c_{1}}{λρ} − \frac{c_{4}}{ρ} − \frac{∂c_{4}}{∂ρ} − \frac{1}{λρ} \frac{∂c_{3}}{∂θ} = 0,

\frac{cos θ}{r} \frac{∂F_{24}}{∂θ} + sin θ\frac{∂F_{24}}{∂r} − \frac{c_{3}}{λρ} − \frac{c_{2}}{ρ} − \frac{∂c_{2}}{∂ρ} − \frac{1}{λρ} \frac{∂c_{1}}{∂θ} = 0,

cos θ\frac{∂c_{1}}{∂r} + sin θ\frac{∂c_{3}}{∂r} − \frac{sin θ}{r} \frac{∂c_{1}}{∂θ} + \frac{cos θ}{r} \frac{∂c_{3}}{∂θ} − \frac{∂F_{13}}{∂ρ} = 0,

cos θ\frac{∂c_{2}}{∂r} + sin θ\frac{∂c_{4}}{∂r} − \frac{sin θ}{r} \frac{∂c_{2}}{∂θ} + \frac{cos θ}{r} \frac{∂c_{4}}{∂θ} + \frac{1}{λρ} \frac{∂F_{13}}{∂θ} = 0.

(46)

A part of the class P_{1,5} is defined by the formula A_{i} = (0, Φ cosh(λφ), 0, −Φ sinh(λφ)), where Φ = Φ(r, ρ, θ) is an arbitrary function. This potential generates the tensor F_{ij} such that

F_{12} = (Φ′_{r} cos(θ − φ) − \frac{1}{r}Φ′_{θ} sin(θ − φ)) cosh λφ, F_{13} = 0,

F_{14} = (−Φ′_{r} cos(θ − φ) + \frac{1}{r}Φ′_{θ} sin(θ − φ)) sinh λφ,

F_{23} = (−Φ′_{r} sin(θ − φ) − \frac{1}{r}Φ′_{θ} cos(θ − φ)) cosh λφ, F_{24} = \frac{1}{λρ}Φ′_{θ},

F_{34} = (−Φ′_{r} sin(θ − φ) − \frac{1}{r}Φ′_{θ} cos(θ − φ)) sinh λφ.

(47)

Proposition 13. If the functions Φ′_{r}, Φ′′_{rr}, Φ′′_{rρ}, and Φ′′_{θθ} are linearly independent, then the tensor (47) defines the Maxwell space that admits the group G_{S} = G_{1,5}.

3. The Maxwell spaces that admit translation groups

In this section, we describe classes of the Maxwell spaces that are invariant with respect to the translation groups whose dimensions are from 2 to 4.

3.1. Two-dimensional subgroups.

3.1.1. The class C_{2,1a}. The algebra L_{2,1a} = L{e_{1}, e_{2}} corresponds to the group G_{2,1a} of translations along the vectors of the Euclidean plane Ox^{1}x^{2}. We have L_{1,1a} ⊂ L_{2,1a}, therefore the class C_{2,1a} is a subclass of the class C_{1,1a}. The Maxwell space of the class C_{2,1a} is defined by the tensor F_{ij} of the form

F_{12} = const, F_{13} = \frac{∂Φ}{∂x^{3}} , F_{14} = \frac{∂Φ}{∂x^{4}} , F_{23} = \frac{∂Ψ}{∂x^{3}} , F_{24} = \frac{∂Ψ}{∂x^{4}} , F_{34} = Θ, (48)

where Φ = Φ(x^{3}, x^{4}), Ψ = Ψ(x^{3}, x^{4}), Θ = Θ(x^{3}, x^{4}) are arbitrary smooth functions.

Proposition 14. The Maxwell space defined by the tensor (48) admits the two-dimensi-onal group G_{S} = G_{2,1a} whenever the partial derivatives ∂_{3}Φ, ∂_{4}Φ, ∂_{3}Ψ, ∂_{4}Ψ are linearly independent and the functions ∂_{3}Θ, ∂_{4}Θ, x^{3}∂_{4}Θ + x^{4}∂_{3}Θ are also linearly independent.

3.1.2. The class C_{2,1b}. The algebra L_{2,1b} = L{e_{2}, e_{4}} corresponds to the group G_{2,1b} of translations along the vectors of the pseudo-Euclidean plane Ox^{2}x^{4}. Since L_{1,1b} ⊂ L_{2,1b}, the class C_{2,1b} is a subclass of the class C_{1,1b}. The Maxwell space of the class C_{2,1b} is defined by the tensor F_{ij} of the form

F_{12} = \frac{∂Φ}{∂x^{1}} , F_{13} = Θ, F_{14} = \frac{∂Ψ}{∂x^{1}} ,
F_{23} = −\frac{∂Φ}{∂x^{3}} , F_{24} = const,  F_{34} = \frac{∂Ψ}{∂x^{3}} ,

(49)

where Φ = Φ(x^{1}, x^{3}), Ψ = Ψ(x^{1}, x^{3}), Θ = Θ(x^{1}, x^{3}) are arbitrary smooth functions.

Proposition 15. The Maxwell space defined by the tensor (49) admits the two-dimensi-onal group G_{S} = G_{2,1b} whenever the partial derivatives ∂_{1}Φ, ∂_{3}Φ, ∂_{1}Ψ, ∂_{3}Ψ are linearly independent and the functions ∂_{1}Θ, ∂_{3}Θ x^{3}∂_{1}Θ − x^{1}∂_{3}Θ are also linearly independent.

3.1.3. The class C_{2,1c}. The algebra L_{2,1c} = L{e_{1}, e_{2} + e_{4}} corresponds to the group G_{2,1c} of translations along the vectors of an isotropic plane. As L_{1,1} ⊂ L_{2,1}, the class C_{2,1c} is a subclass of the class C_{1,1c}. The Maxwell space of the class C_{2,1c} is defined by the tensor F_{ij} of the form

\begin{aligned}F_{12} & = −\frac{∂Φ}{∂v^{4}} , & F_{13} & = −\frac{∂Φ}{∂v^{3}} , & F_{14} & = \frac{∂Φ}{∂v^{4}} , \\ F_{23} & = Θ, & F_{24} & = \frac{∂Ψ}{∂v^{4}} , & F_{34} & = Θ + \frac{∂Ψ}{∂v^{3}} , \end{aligned}  (50)

where C = const and Φ = Φ(v^{3}, v^{4}), Ψ = Ψ(v^{3}, v^{4}), and Θ = Θ(v^{3}, v^{4}) are arbitrary smooth functions (v^{3} = x^{3}, v^{4} = x^{2} − x^{4}).

Set C = 0 and Θ = 0 in (50):

F_{12} = −\frac{∂Φ}{∂v^{4}} , F_{13} = −\frac{∂Φ}{∂v^{3}} , F_{14} = \frac{∂Φ}{∂v^{4}} , F_{23} = 0, F_{24} = \frac{∂Ψ}{∂v^{4}} , F_{34} = \frac{∂Ψ}{∂v^{3}} .  (51)

Proposition 16. The Maxwell space defined by the tensor (50) admits the two-dimen-sional group G_{S} = G_{2,1c} whenever the partial derivatives ∂Φ/∂v^{3}, ∂Φ/∂v^{4}, ∂Ψ/∂v^{3}, ∂Ψ/∂v^{4} are linearly independent and the functions Φ_{34} = ∂^{2}Φ/∂v^{3}∂v^{4}, Φ_{44} = ∂^{2}Φ/∂v^{4}∂v^{4}, x^{3}Φ_{44} − x^{2}Φ_{34} are also linearly independent.

3.2. Three-dimensional subgroups.

3.2.1. The class C_{3,1a}. The algebra L_{3,1a} = L{e_{1}, e_{2}, e_{3}} corresponds to the group G_{3,1a} of translations along the vectors of three-dimensional Euclidean space Ox^{1}x^{2}x^{3}. The class C_{3,1a} is a subclass of C_{2,1a}. The Maxwell space of the class C_{3,1a} is defined by the tensor F_{ij} of the form

F_{12} = C_{3}, F_{23} = C_{2}, F_{13} = C_{1},

F_{14} = φ′(x^{4}), F_{24} = ψ′(x^{4}), F_{34} = χ(x^{4}). (52)

where C_{k} = const and φ(x^{4}), ψ(x^{4}), χ(x^{4}) are arbitrary smooth functions (the prime denotes differentiation). In particular, the homogeneous magnetic field crossed with the electric field that depends only on time belongs to the class C_{3,1a}.

Proposition 17. If the functions φ(x^{4}), φ′(x^{4}), and ψ(x^{4}) are linearly independent and χ(x^{4}) ≠ 0, then the Maxwell space defined by the tensor (52) admits the three-di-mensional group G_{S} = G_{3,1a}.

3.2.2. The class C_{3,1b}. The algebra L_{3,1b} = L{e_{1}, e_{2}, e_{4}} corresponds to the group G_{3,1b} of translations along the vectors of three-dimensional pseudo-Euclidean space Ox^{1}x^{2}x^{4}. The class C_{3,1b} is a subclass of C_{2,1b}. The Maxwell space of the class C_{3,1b} is defined by the tensor F_{ij} of the form

F_{12} = C_{1}, F_{13} = φ(x^{3}), F_{14} = C_{2},

F_{23} = ψ(x^{3}), F_{24} = C_{3}, F_{34} = χ(x^{3}), (53)

where C_{k} = const and φ(x^{3}), ψ(x^{3}), and χ(x^{3}) are arbitrary smooth functions.

Proposition 18. If the functions φ(x^{3}), φ′(x^{3}), ψ(x^{3}), and χ(x^{3}) are linearly independent, then the Maxwell space defined by the tensor (53) admits the three-dimensional group G_{S} = G_{3,1b}.

For example, this condition holds for the functions φ = x^{3}, ψ = sin x^{3}, χ = cos x^{3}.

3.2.3. The class C_{3,1c}. The algebra L_{3,1c} = L{e_{1}, e_{3}, e_{2} + e_{4}} corresponds to the group G_{3,1c} of translations along the vectors of a three-dimensional isotropic space. The class C_{3,1c} is a subclass of C_{2,1c}. The Maxwell space of the class C_{3,1c} is defined by the tensor F_{ij} of the form

F_{ij} = F_{ij}(v^{4}) (ij = 12, 23, 24), F_{13} = C_{1}, F_{14} = C_{2} − F_{12}, F_{34} = C_{3} + F_{23}, (54)

where C_{k} = const and F_{ij} = F_{ij}(v^{4}) = F_{ij}(x^{2} − x^{4}) are arbitrary smooth functions.

Proposition 19. If the functions F_{ij} = F_{ij}(v^{4}) (ij = 12, 23, 24) are linearly independent and F_{24} ≠ const, then the Maxwell space defined by the tensor (54) admits the three-dimensional group G_{S} = G_{3,1c}.

3.3. Four-dimensional subgroup. The algebra L_{4,1} = L{e_{1}, e_{2}, e_{3}, e_{4}} corresponds to the group G_{4,1} of translations of the Minkowski space R_{1}^{4}. The Maxwell space of the class C_{4,1} is defined by the constant tensor F_{ij}. Therefore, C_{4,1} is the class of homogeneous Maxwell spaces.

Remark 2. It is easily proved that the group G_{S} is six-dimensional for each homogeneous Maxwell’s space [20]. Hence, there are no Maxwell’s spaces with the symmetry group G_{S} = G_{4,1}.

4. The classes of the static Maxwell spaces

In this section, we describe classes of the static Maxwell spaces which were not discussed in sections 2 and 3. See details in [21].

4.1. Two-dimensional subgroup. The algebra L_{2,3} = L{e_{13} + λe_{2}, e_{4}} corresponds to the group G_{2,3} generated by elliptic helices with a space-like axis and by translations along a time-like straight line. The Maxwell space of the class C_{2,3} is defined by the tensor F_{ij} of the form

F_{12} = c_{1} cos φ + c_{2} sin φ, F_{13} = F_{13}(r, x̃^{2}), F_{14} = c_{3} cos φ + c_{4} sin φ,

F_{23} = c_{1} sin φ − c_{2} cos φ, F_{24} = F_{24}(r, x̃^{2}), F_{34} = −c_{3} sin φ + c_{4} cos φ, (55)

where the functions F_{13}(r, x̃^{2}), F_{24}(r, x̃^{2}) and c_{i} = c_{i}(r, x̃^{2}) satisfy the equations

\begin{matrix}\frac{∂c_{1}}{∂r} + \frac{c_{1}}{r} + \frac{λ}{r} \frac{∂c_{2}}{∂x̃^{2}} − \frac{∂F_{13}}{∂x̃^{2}} = 0, \frac{∂F_{24}}{∂r} − \frac{∂c_{4}}{∂x̃^{2}} = 0, \\ \frac{∂c_{3}}{∂r} + \frac{c_{3}}{r} + \frac{λ}{r} \frac{∂c_{4}}{∂x̃^{2}} = 0,  \frac{λ}{r} \frac{∂F_{24}}{∂x̃^{2}} + \frac{∂c_{3}}{∂x̃^{2}} = 0; \end{matrix}  (56)

the coordinates {x^{i}} and {r, x̃^{2}, φ, x̃^{4}} are related by (12) for μ = 0:

x^{1} = r sin φ, x^{2} = λφ + x̃^{2}, x^{3} = r cos φ, x^{4} = x̃^{4}.  (57)

The class of the potentials P_{2,3} invariant with respect to the group G_{2,3} consists of the fields

A_{1} = b_{1} cos φ + b_{2} sin φ, A_{2} = A_{2}(r, x̃^{2}),

A_{3} = −b_{1} sin φ + b_{2} cos φ, A_{4} = A_{4}(r, x̃^{2}), (58)

where b_{k} = b_{k}(r, x̃^{2}), A_{2}(r, x̃^{2}), and A_{4}(r, x̃^{2}) are arbitrary smooth functions. Setting A_{1} = A_{2} = A_{3} = 0, A_{4} = Φ(r, x̃^{2}) ≡ Φ(t_{1}, t_{2}), we obtain the following set of electrostatic fields

F_{12} = F_{13} = F_{23} = 0, F_{14} = Φ_{1} sin φ − \frac{λΦ_{2}}{r} cos φ,

F_{24} = Φ_{2}, F_{34} = Φ_{1} cos φ + \frac{λΦ_{2}}{r} sin φ,

(59)

where Φ_{α} = ∂Φ/∂t_{α}. Let Φ_{αβ} = ∂^{2}Φ/∂t_{α}∂t_{β}.

Proposition 20. If the partial derivatives Φ_{1}, Φ_{2}, Φ_{11}, Φ_{12} and Φ_{22} are linearly independent, then the Maxwell space defined by the tensor (59) admits the two-dimensional group G_{S} = G_{2,3}.

In particular, this condition holds for the function Φ(t_{1}, t_{2}) = t_{1}^{3} + t_{2}^{3} + t_{1}^{2} + t_{1}t_{2} + t_{2}^{2}.

4.2. Three-dimensional subgroup. The algebra L_{3,6} = L{e_{24} + λe_{3}, e_{2}, e_{4}} corresponds to the group G_{3,6} generated by hyperbolic helices and translations along the vectors of the pseudo-Euclidean plane. The Maxwell space of the class C_{3,6} is defined by the tensor F_{ij} as follows: a) for λ ≠ 0 (class C_{3,6a})

F_{12} = −λc′_{1}(x^{1}) cosh \frac{x^{3}}{λ} − λc′_{2}(x^{1}) sinh \frac{x^{3}}{λ} , F_{13} = F_{13}(x^{1}),

F_{14} = λc′_{1}(x^{1}) sinh \frac{x^{3}}{λ} + λc′_{2}(x^{1}) cosh \frac{x^{3}}{λ} , F_{24} = const,

F_{23} = c_{1}(x^{1}) sinh \frac{x^{3}}{λ} + c_{2}(x^{1}) cosh \frac{x^{3}}{λ} ,

F_{34} = c_{1}(x^{1}) cosh \frac{x^{3}}{λ} + c_{2}(x^{1}) sinh \frac{x^{3}}{λ} ,

(60)

where c_{1}(x^{1}) and c_{2}(x^{1}) are arbitrary functions (prime denotes differentiation); b) For λ = 0 (class C_{3,6b})

F_{12} = F_{14} = F_{23} = F_{34} = 0, F_{13} = F_{13}(x^{1}, x^{3}), F_{24} = const.  (61)

Putting c_{2} = F_{13} = F_{24} = 0 in (60), we get

F_{12} = −λc′_{1}(x^{1}) cosh \frac{x^{3}}{λ} , F_{13} = 0, F_{14} = λc′_{1}(x^{1}) sinh \frac{x^{3}}{λ} ,

F_{23} = c_{1}(x^{1}) sinh \frac{x^{3}}{λ} , F_{24} = 0, F_{34} = c_{1}(x^{1}) cosh \frac{x^{3}}{λ} .

(62)

Proposition 21. If the functions c_{1}(x^{1}) and c′_{1}(x^{1}) are linearly independent, then the Maxwell space defined by the tensor (62) admits the three-dimensional group G_{S} = G_{3,6a}.

Proposition 22. If the functions ∂_{1}F_{13}, ∂_{3}F_{13}, and x^{3}∂_{1}F_{13} − x^{1}∂_{3}F_{13} are linearly independent, then the Maxwell space defined by the tensor (61) admits the three-dimen-sional group G_{S} = G_{3,6b}.

4.3. Four-dimensional subgroups.

4.3.1. The class C_{4,3}. The algebra L_{4,3} = L{e_{13} + λe_{2}, e_{1}, e_{3}, e_{4}} corresponds to the group G_{4,3} generated by elliptic helices with a space-like axis and translations along the vectors of the pseudo-Euclidean hyperplane. The Maxwell space of the class C_{4,3} is defined by the tensor F_{ij} as follows:

a) for λ ≠ 0 (class C_{4,3a})

F_{12} = a_{1} sin \frac{x^{2}}{λ} − a_{2} cos \frac{x^{2}}{λ} , F_{32} = a_{1} cos \frac{x^{2}}{λ} + a_{2} sin \frac{x^{2}}{λ} ,

F_{14} = F_{34} = 0, F_{13} = a_{3}, F_{24} = a_{4} (a_{i} = const);

(63)

b) for λ = 0 (class C_{4,3b})

F_{12} = F_{14} = F_{23} = F_{34} = 0, F_{13} = Φ(x^{2}), F_{24} = Ψ(x^{2}),  (64)

where Φ(x^{2}) � Ψ(x^{2}) are arbitrary functions.

Proposition 23. If a_{1} ≠ 0 or a_{2} ≠ 0, then the Maxwell space defined by the tensor (63) admits the four-dimensional group G_{S} = G_{4,3a}.

Proposition 24. If the functions Φ(x^{2}) and Ψ(x^{2}) are linearly independent, then the Maxwell space defined by the tensor (64) admits the four-dimensional group G_{S} = G_{4,3b}.

4.3.2. The class C_{4,6}. The algebra L_{4,6} = L{e_{24} + λe_{3}, e_{1}, e_{2}, e_{4}} corresponds to the group G_{4,6} generated by hyperbolic helices and translations along the vectors of the pseudo-Euclidean hyperplane. The Maxwell space of the class C_{4,6} is defined by the tensor F_{ij} of the form:

a) for λ ≠ 0 (class C_{4,6a})

F_{23} = a_{1} cosh \frac{x^{3}}{λ} + a_{2} sinh \frac{x^{3}}{λ} , F_{34} = a_{1} sinh \frac{x^{3}}{λ} + a_{2} cosh \frac{x^{3}}{λ} ,

F_{12} = F_{14} = 0, F_{24} = a_{3}, F_{13} = a_{4} (a_{i} = const);

(65)

b) for λ = 0 (class C_{4,6b})

F_{12} = F_{14} = F_{23} = F_{34} = 0, F_{13} = F_{13}(x^{3}), F_{24} = const.  (66)

Proposition 25. If a_{1} ≠ 0 or a_{2} ≠ 0, then the Maxwell space defined by the tensor (65) admits the four-dimensional group G_{S} = G_{4,6a}.

Proposition 26. If F_{13}(x^{3}) ≠ const, then the Maxwell space defined by the tensor (66) admits the four-dimensional group G_{S} = G_{4,6b}.

4.3.3. The class C_{4,8}. The algebra L_{4,8} = L{e_{12} − e_{14} + λe_{3}, e_{1}, e_{2}, e_{4}} corresponds to the group G_{4,8} generated by parabolic helices and translations along the vectors of the pseudo-Euclidean hyperplane. The Maxwell space of the class C_{4,8} is defined by the tensor F_{ij} of the form:

a) for λ = 0 (class C_{4,8a})

F_{12} = F_{14} = const, F_{13} = F_{24} = 0, F_{23} = −F_{34}(x^{3});  (67)

b) for λ ≠ 0 (class C_{4,8b})

\begin{matrix}F_{12} = F_{14} = C_{1}, F_{24} = 0, F_{13} = \frac{C_{2}}{λ} x^{3} + C_{3}, \\ F_{23} = \frac{C_{2}}{2λ^{2}} (x^{3})^{2} + \frac{C_{3}}{λ} x^{3} + C_{4}, F_{34} = −F_{23} − C_{2}, \end{matrix}  (68)

where C_{1}, …, C_{4} are arbitrary constants.

Proposition 27. If F_{23} ≠ const, then the Maxwell space defined by the tensor (67) admits the four-dimensional group G_{S} = G_{4,8a}.

Proposition 28. If C_{1} ≠ 0 and C_{2} ≠ 0, then the Maxwell space defined by the tensor (68) admits the four-dimensional group G_{S} = G_{4,8b}.

4.3.4. The class C_{4,18}. The algebra L_{4,18} = L{e_{12}, e_{13}, e_{23}, e_{4}} corresponds to the group G_{4,18} generated by all rotations around the origin O in Euclidean space Ox^{1}x^{2}x^{3} and translations along the time axis Ox^{4}. The Maxwell space of the class C_{4,18} is defined by the tensor F_{ij} of the form:

\begin{matrix}F_{12} = −Ax^{3}/r^{3}, F_{13} = Ax^{2}/r^{3}, F_{23} = −Ax^{1}/r^{3}, \\ F_{14} = x^{1}C(r), F_{24} = x^{2}C(r), F_{34} = x^{3}C(r), \end{matrix}  (69)

where A = const, C(r) an arbitrary function and {r, φ, θ, x^{4}} are “spherical” coordinates

x^{1} = r cos φ cos θ, x^{2} = r sin φ cos θ, x^{3} = r sin θ.  (70)

Example 2. If A = 0 and C(r) = K/r^{3} (K = const), then (69) is the Coulomb field. It admits the four-dimensional group G_{S} = G_{4,18}.

4.4. Five- and six-dimensional subgroups.

4.4.1. For each algebra L_{5,1} = L{e_{24}, e_{1}, e_{2}, e_{3}, e_{4}}, L_{5,2} = L{e_{13} + λe_{24}, e_{1}, e_{2}, e_{3}, e_{4}}, and L_{6,2} = L{e_{13}, e_{24}, e_{1}, e_{2}, e_{3}, e_{4}}, there is the class C_{6,2} of the homogeneous Maxwell spaces defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{23} = F_{34} = 0, F_{13} = C_{1}, F_{24} = C_{2},  (71)

where C_{1} and C_{2} are arbitrary constants.

Proposition 29. If C_{1} ≠ 0 or C_{2} ≠ 0, then the Maxwell space defined by the tensor (71) admits the six-dimensional group G_{S} = G_{6,2} which corresponds to the algebra L_{6,2}. There are no Maxwell’s spaces with the symmetry groups G_{S} = G_{5,1} and G_{S} = G_{5,2}.

4.4.2. For each algebra

L_{5,3} = L{e_{12} − e_{14}, e_{1}, e_{2}, e_{3}, e_{4}} and L_{6,3} = L{e_{12} − e_{14}, e_{23} + e_{34}, e_{1}, e_{2}, e_{3}, e_{4}}

there is the class C_{6,3} of the homogeneous Maxwell spaces defined by the tensor F_{ij} of the form

F_{12} = F_{14} = C_{1}, F_{23} = −F_{34} = C_{2}, F_{13} = F_{24} = 0,  (72)

where C_{1} and C_{2} are arbitrary constants.

Proposition 30. If C_{1} ≠ 0 or C_{2} ≠ 0, then the Maxwell space defined by the tensor (72) admits the six-dimensional group G_{S} = G_{6,3} which corresponds to the algebra L_{6,3}. There are no Maxwell’s spaces with symmetry group G_{S} = G_{5,3}.

4.4.3. The algebra L_{5,6} = L{e_{12} − e_{14}, e_{24} + λe_{3}, e_{1}, e_{2}, e_{4}} corresponds to the group G_{5,6} generated by parabolic rotations, hyperbolic helices, and translations along the vectors of the pseudo-Euclidean hyperplane. Since L_{4,6} ⊂ L_{5,6}, then C_{5,6} ⊂ C_{4,6}. The Maxwell space of the class C_{5,6} is defined by the tensor F_{ij} of the form:

F_{12} = F_{13} = F_{14} = F_{24} = 0, F_{23} = −F_{34} = K e^{−x^{3}/λ} (K = const).  (73)

For λ = 0 this class is empty.

Proposition 31. For K ≠ 0 the Maxwell space defined by the tensor (73), admits the five-dimensional group G_{S} = G_{5,6}.

4.4.4. For the algebras

L_{6,4} = L{e_{12} − e_{14}, e_{24}, e_{1}, e_{2}, e_{3}, e_{4}} and

L_{6,9} = L{e_{12}, e_{14}, e_{24}, e_{1}, e_{2}, e_{4}},

the corresponding classes of Maxwell spaces are empty.

5. The Maxwell spaces that admit elliptic helices

In this section, we describe classes of the Maxwell spaces that admit elliptic helices and were not included in the previous sections. See details in [21].

5.1. Two-dimensional subgroups.

5.1.1. The class C_{2,2}. The algebra L_{2,2} = L{e_{13}+μe_{4}, e_{2}} corresponds to the group G_{2,2} generated by elliptic helices with a time-like axis and by translations along a space-like straight line. Since L_{1,2b} ⊂ L_{2,2}, then C_{2,2} ⊂ C_{1,2b}. The Maxwell space of the class C_{2,2} is defined by the tensor F_{ij} of the form

F_{12} = c_{1} cos φ + c_{2} sin φ, F_{13} = F_{13}(r, x̃^{4}), F_{14} = c_{3} cos φ + c_{4} sin φ,

F_{23} = c_{1} sin φ − c_{2} cos φ, F_{24} = F_{24}(r, x̃^{4}), F_{34} = −c_{3} sin φ + c_{4} cos φ, (74)

where c_{i} = c_{i}(r, x̃^{4}) are the functions that satisfy to the following system of equations

\begin{matrix}\frac{∂c_{1}}{∂r} + \frac{c_{1}}{r} + \frac{μ}{r} \frac{∂c_{2}}{∂x̃^{4}} = 0,  \frac{∂F_{24}}{∂r} + \frac{∂c_{2}}{∂x̃^{4}} = 0, \\ \frac{∂c_{3}}{∂r} + \frac{c_{3}}{r} + \frac{μ}{r} \frac{∂c_{4}}{∂x̃^{4}} − \frac{∂F_{13}}{∂x̃^{4}} = 0, \frac{μ}{r} \frac{∂F_{24}}{∂x̃^{4}} − \frac{∂c_{1}}{∂x̃^{4}} = 0; \end{matrix}  (75)

the relation of the coordinates {x^{i}} and {x̃^{i}} = {r, x̃^{2}, φ, x̃^{4}} is defined by (12) for λ = 0:

x^{1} = r sin φ, x^{2} = x̃^{2}, x^{3} = r cos φ, x^{4} = μφ + x̃^{4}.  (76)

For the potential A_{i} = (0, 0, 0, Φ), Φ = Φ(r, x̃^{4}), we get instead of (20):

F_{14} = Φ_{r} sin φ − \frac{μ}{r}Φ_{4} cos φ, F_{12} = F_{13} = F_{23} = F_{24} = 0,

F_{34} = Φ_{r} cos φ + \frac{μ}{r}Φ_{4} sin φ (Φ_{r} = \frac{∂Φ}{∂r}, Φ_{4} = \frac{∂Φ}{∂x̃^{4}} ) .

(77)

Proposition 32. If ∂^{2}Φ/∂r∂x̃^{4} ≠ 0, then the Maxwell space defined by the tensor (77) admits the two-dimensional group G_{S} = G_{2,2}.

5.1.2. The class C_{2,4}. The algebra L_{2,4} = L{e_{13} + λe_{2}, e_{2} + e_{4}} corresponds to the group G_{2,4} generated by elliptic helices with a space-like axis and by translations along an isotropic straight line. The class C_{2,4} is an intersection of the classes C_{1,2a} and C_{1,1c}. The Maxwell space of the class C_{2,4} is defined by the tensor F_{ij} of the form

F_{12} = c_{1} cos φ + c_{2} sin φ, F_{13} = F_{13}(r, x̃^{2} − x̃^{4}),
F_{14} = c_{3} cos φ + c_{4} sin φ, F_{23} = c_{1} sin φ − c_{2} cos φ,
F_{24} = F_{24}(r, x̃^{2} − x̃^{4}), F_{34} = −c_{3} sin φ + c_{4} cos φ,

(78)

where the functions c_{i} = c_{i}(r, x̃^{2} − x̃^{4}), i = 1, …, 4, satisfy to (17) (the change of coordinates is defined by (12) for μ = 0):

x^{1} = r sin φ, x^{2} = λφ + x̃^{2}, x^{3} = r cos φ, x^{4} = x̃^{4}.  (79)

For the potential A_{i} = (0, 0, 0, Φ), Φ = Φ(r, x̃^{2} − x̃^{4}) = Φ(r, u), we get instead of (18):

F_{12} = F_{13} = 0, F_{14} = Φ_{r} sin φ − \frac{λ}{r}Φ_{u} cos φ,

F_{23} = 0, F_{24} = Φ_{u}, F_{34} = Φ_{r} cos φ + \frac{λ}{r}Φ_{u} sin φ.

(80)

Proposition 33. If ∂^{2}Φ/∂r∂u ≠ 0, then the Maxwell space defined by the tensor (80) admits the two-dimensional group G_{S} = G_{2,4}.

5.2. Three-dimensional subgroups.

5.2.1. The class C_{3,2}. The algebra L_{3,2} = L{e_{13}+λe_{2}, e_{1}, e_{3}} (λ ≠ 0) corresponds to the group G_{3,2} generated by elliptic helices with a space-like axis and by translations along the vectors of the two-dimensional Euclidian plane. Since L_{1,2a} ⊂ L_{3,2} and L_{1,1a} ⊂ L_{3,2}, then C_{3,2} ⊂ C_{1,2a} and C_{3,2} ⊂ C_{1,1a}. The Maxwell space of the class C_{3,2} is defined by the tensor F_{ij} of the form

F_{12} = a_{1}(x^{4}) sin \frac{x^{2}}{λ} − a_{2}(x^{4}) cos \frac{x^{2}}{λ} , F_{13} = const,

F_{23} = −a_{1}(x^{4}) cos \frac{x^{2}}{λ} − a_{2}(x^{4}) sin \frac{x^{2}}{λ} , F_{24} = F_{24}(x^{4}),

F_{14} = −λa′_{1}(x^{4}) cos \frac{x^{2}}{λ} − λa′_{2}(x^{4}) sin \frac{x^{2}}{λ} ,

F_{34} = λa′_{1}(x^{4}) sin \frac{x^{2}}{λ} − λa′_{2}(x^{4}) cos \frac{x^{2}}{λ} ,

(81)

where a_{k} = a_{k}(x^{4}) are arbitrary smooth functions.

Proposition 34. The Maxwell space defined by the tensor (81) admits the three-di-mensional group G_{S} = G_{3,2} whenever one of the following conditions is satisfied:

1) the functions a_{1}(x^{4}) and a′_{1}(x^{4}) are linearly independent,

2) the functions a_{2}(x^{4}) and a′_{2}(x^{4}) are linearly independent.

5.2.2. The class C_{3,3}. The algebra L_{3,3} = L{e_{13}+μe_{4}, e_{1}, e_{3}} (μ ≠ 0) corresponds to the group G_{3,3} generated by elliptic helices with a time-like axis and by translations along the vectors of the two-dimensional Euclidian plane. Since L_{1,2b} ⊂ L_{3,3} and L_{1,1a} ⊂ L_{3,3}, then C_{3,3} ⊂ C_{1,2b} and C_{3,3} ⊂ C_{1,1a}. The Maxwell space of the class C_{3,3} is defined by the tensor F_{ij} of the form

F_{12} = μa′_{1}(x^{2}) sin \frac{x^{4}}{μ} − μa′_{2}(x^{2}) cos \frac{x^{4}}{μ} , F_{13} = const,

F_{23} = −μa′_{1}(x^{2}) cos \frac{x^{4}}{μ} − μa′_{2}(x^{2}) sin \frac{x^{4}}{μ} , F_{24} = F_{24}(x^{2}),

F_{14} = a_{1}(x^{2}) cos \frac{x^{4}}{μ} + a_{2}(x^{2}) sin \frac{x^{4}}{μ} ,

F_{34} = −a_{1}(x^{2}) sin \frac{x^{4}}{μ} + a_{2}(x^{2}) cos \frac{x^{4}}{μ} ,

(82)

where a_{1}(x^{2}) and a_{2}(x^{2}) are arbitrary smooth functions.

Proposition 35. The Maxwell space defined by the tensor (82) admits the three-di-mensional group G_{S} = G_{3,3}, whenever one of the following conditions is satisfied:

1) the functions a_{1}(x^{2}) and a′_{1}(x^{2}) are linearly independent,

2) the functions a_{2}(x^{2}) and a′_{2}(x^{2}) are linearly independent.

5.2.3. The class C_{3,4a}. The algebra L_{3,4a} = L{e_{13} + λ(e_{2} + e_{4}), e_{1}, e_{3}} (λ ≠ 0) corresponds to the group G_{3,4a} generated by elliptic helices with an isotropic axis and by translations along the vectors of the two-dimensional Euclidian plane. Since L_{1,2c} ⊂ L_{3,4a} and L_{1,1a} ⊂ L_{3,4a}, then C_{3,4a} ⊂ C_{1,2c} and C_{3,4a} ⊂ C_{1,1a}. The Maxwell space of the class C_{3,4a} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = a_{1} sin \frac{x^{2} + x^{4}}{2λ} − a_{2} cos \frac{x^{2} + x^{4}}{2λ} , F_{13} = const,

F_{34} = −F_{23} = a_{1} cos \frac{x^{2} + x^{4}}{2λ} + a_{2} sin \frac{x^{2} + x^{4}}{2λ} , F_{24} = Φ(x^{2} − x^{4}),

(83)

where a_{1}, a_{2} = const and Φ(u) is an arbitrary function of one variable.

Proposition 36. The Maxwell space defined by the tensor (83) admits the three-di-mensional group G_{S} = G_{3,4a}, whenever one of the following conditions is satisfied:

1) a_{1} ≠ 0 and Φ(u) ≠ const,

2) a_{2} ≠ 0 and Φ(u) ≠ const.

5.2.4. The class C_{3,4b}. The algebra L_{3,4b} = L{e_{13}, e_{1}, e_{3}} corresponds to the group G_{3,4b} of motions of the two-dimensional Euclidian plane Ox^{1}x^{3}. Since L_{1,2d} ⊂ L_{3,4b} and L_{1,1a} ⊂ L_{3,4b}, then C_{3,4b} ⊂ C_{1,2d} and C_{3,4b} ⊂ C_{1,1a}. The Maxwell space of the class C_{3,4b} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{23} = F_{34} = 0, F_{13} = Φ(x^{2}, x^{4}), F_{24} = Ψ(x^{2}, x^{4}),  (84)

where Φ(x^{2}, x^{4}) and Ψ(x^{2}, x^{4}) are arbitrary smooth functions.

Proposition 37. If the functions Φ(x^{2}, x^{4}), Ψ(x^{2}, x^{4}) are linearly independent and the partial derivatives ∂Φ/∂x^{2}, ∂Φ/∂x^{4} (or ∂Ψ/∂x^{2}, and ∂Ψ/∂x^{4}) are also linearly independent, then the Maxwell space defined by the tensor (84) admits the three-dimen-sional group G_{S} = G_{3,4b}.

5.3. Four-dimensional subgroups.

5.3.1. The class C_{4,2a}. The algebra L_{4,2a} = L{e_{13} + μe_{4}, e_{1}, e_{2}, e_{3}} (μ ≠ 0) corresponds to the group G_{4,2a} generated by elliptic helices with a time-like axis and by translations along the vectors of the Euclidian hyperplane. Since L_{3,3a} ⊂ L_{4,2a}, then C_{4,2a} ⊂ C_{3,3a}. The Maxwell space of the class C_{4,2a} is defined by the tensor F_{ij} of the form

F_{14} = c_{1} cos \frac{x^{4}}{μ} + c_{2} sin \frac{x^{4}}{μ} , F_{34} = −c_{1} sin \frac{x^{4}}{μ} + c_{2} cos \frac{x^{4}}{μ} ,

F_{12} = F_{23} = 0, F_{13} = c_{3}, F_{24} = c_{4} (c_{k} = const).

(85)

Proposition 38. If c_{1} ≠ 0 or c_{2} ≠ 0, then the Maxwell space defined by the tensor (85) admits the four-dimensional group G_{S} = G_{4,2a}.

5.3.2. The class C_{4,2b}. The algebra L_{4,2b} = L{e_{13}, e_{1}, e_{2}, e_{3}} corresponds to the group G_{4,2b} generated by rotations in the plane Ox^{1}x^{3} and by translations along the vectors of the Euclidian hyperplane. Since L_{3,4b} ⊂ L_{4,2b}, then C_{4,2b} ⊂ C_{3,4b}. The Maxwell space of the class C_{4,2b} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{23} = F_{34} = 0,

F_{13} = C = const, F_{24} = Φ(x^{4}), (86)

where Φ(x^{4}) is an arbitrary smooth function.

Proposition 39. If ≠ 0 and Φ(x^{4}) ≠ const, then the Maxwell space defined by the tensor (86) admits the four-dimensional group G_{S} = G_{4,2b}.

5.3.3. The class C_{4,4a}. The algebra L_{4,4a} = L{e_{13} + λe_{2}, e_{1}, e_{3}, e_{2} + e_{4}} (λ ≠ 0) corresponds to the group G_{4,4a} generated by elliptic helices with a space-like axis and by translations along the vectors of the isotropic hyperplane. Since L_{3,2} ⊂ L_{4,4a}, then C_{4,4a} ⊂ C_{3,2}. The Maxwell space of the class C_{4,2a} is defined by the tensor F_{ij} of the form

F_{12} = −F_{14} = b_{1} cos \frac{x^{2} − x^{4}}{λ} − b_{2} sin \frac{x^{2} − x^{4}}{λ} , F_{13} = b_{3},

F_{23} = F_{34} = b_{1} sin \frac{x^{2} − x^{4}}{λ} + b_{2} cos \frac{x^{2} − x^{4}}{λ} , F_{24} = b_{4} (b_{i} = const).

(87)

Proposition 40. The Maxwell space defined by the tensor (87) admits the four-dimen-sional group G_{S} = G_{4,4a}, whenever the following conditions are satisfied

1) b_{1} ≠ 0 or b_{2} ≠ 0,

2) b_{3} ≠ 0 or b_{4} ≠ 0.

5.3.4. The class C_{4,4b}. The algebra L_{4,4b} = L{e_{13}, e_{1}, e_{3}, e_{2} + e_{4}} corresponds to the group G_{4,4b} generated by rotations in the plane Ox^{1}x^{3} and by translations along the vectors of the isotropic hyperplane. Since L_{3,4b} ⊂ L_{4,4b}, then C_{4,4b} ⊂ C_{3,4b}. The Maxwell space of the class C_{4,2b} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{23} = F_{34} = 0, F_{13} = Φ(x^{2} − x^{4}), F_{24} = Ψ(x^{2} − x^{4}),  (88)

where Φ(u) and Ψ(u) are arbitrary functions of one variable.

Proposition 41. The Maxwell space defined by the tensor (88) admits the four-dimen-sional group G_{S} = G_{4,4b}, whenever two following conditions are satisfied

1) the functions Φ(u) and Ψ(u) are linearly independent,

2) Φ(u) ≠ const or Ψ(u) ≠ const.

5.4. Five- and six-dimensional subgroups.

5.4.1. For the algebra L_{5,4} = L{e_{13}, e_{24}, e_{1}, e_{3}, e_{2} + e_{4}} we have the class C_{6,2} of homogeneous Maxwell spaces defined by the tensor F_{ij} of the form (71).

Proposition 42. There are no Maxwell’s spaces with the symmetry group G_{S} = G_{5,4}.

5.4.2. The classes of the Maxwell spaces that correspond to the algebras

L_{6,5} = L{e_{12} − e_{14}, e_{23} + e_{34}, e_{13} + λe_{2}, e_{1}, e_{3}, e_{2} − e_{4}}

and

L_{6,8} = L{e_{12}, e_{13}, e_{23}, e_{1}, e_{2}, e_{3}},

are empty.

6. The Maxwell spaces that admit hyperbolic helices

In this section, we describe classes of the Maxwell spaces that admit hyperbolic helices and were not included in the previous sections. See details in [21].

6.1. Two-dimensional subgroups.

6.1.1. The class C_{2,5}. The algebra L_{2,5} = L{e_{24} + λe_{3}, e_{1}} corresponds to the group G_{2,5} generated by hyperbolic helices and by translations along the space-like straight line Ox^{1}. The Maxwell space of the class C_{2,5} is defined by the tensor F_{ij} of the form

F_{12} = c_{1} cosh φ + c_{2} sinh φ, F_{13} = F_{13}(r, x̃^{3}),
F_{14} = −c_{1} sinh φ − c_{2} cosh φ, F_{23} = c_{3} cosh φ + c_{4} sinh φ,
F_{24} = F_{24}(r, x̃^{3}), F_{34} = c_{3} sinh φ + c_{4} cosh φ,

(89)

where the functions c_{i} = c_{i}(r, x̃^{3}) satisfy the equations

\begin{matrix}\frac{∂c_{2}}{∂r} + \frac{c_{2}}{r} − \frac{λ}{r} \frac{∂c_{1}}{∂x̃^{3}} = 0,  \frac{∂c_{1}}{∂x̃^{3}} − \frac{∂F_{13}}{∂r} = 0, \\ \frac{∂c_{4}}{∂r} + \frac{c_{4}}{r} − \frac{λ}{r} \frac{∂c_{3}}{∂x̃^{3}} − \frac{∂F_{24}}{∂x̃^{3}} = 0, \frac{λ}{r} \frac{∂F_{13}}{∂x̃^{3}} − \frac{∂c_{2}}{∂x̃^{3}} = 0, \end{matrix}  (90)

and the transformation of coordinates is defined by formulae

x^{1} = x̃^{1}, x^{2} = r cosh φ, x^{3} = λφ + x̃^{3}, x^{4} = r sinh φ.  (91)

The class P_{2,5} of the potentials A_{i} that are invariant with respect to the group G_{2,5} consists of the following fields [23]

A_{1} = A_{1}(r, x̃^{3}), A_{2} = C_{1} cosh φ + C_{2} sinh φ,

A_{3} = A_{3}(r, x̃^{3}), A_{4} = −C_{1} sinh φ − C_{2} cosh φ, (92)

where C_{i} = C_{i}(r, x̃^{3}). For the field A_{i} = (−Φ, 0, 0, 0), Φ = Φ(x̃^{3}), we obtain

F_{12} = \frac{λ}{r}Φ′ sinh φ, F_{13} = Φ′, F_{14} = −\frac{λ}{r}Φ′ cosh φ, F_{23} = F_{24} = F_{34} = 0,  (93)

where Φ′ = dΦ/dx̃^{3}.

Proposition 43. If Φ′′(x̃^{3}) ≠ 0, then the Maxwell space defined by the tensor (93) admits the two-dimensional group G_{S} = G_{2,5}.

6.1.2. The class C_{2,6}. The algebra L_{2,6} = L{e_{24} +λe_{3}, e_{2} −e_{4}} corresponds to the group G_{2,6} generated by hyperbolic helices and by translations along an isotropic straight line. The Maxwell space of the class C_{2,6} is defined by the tensor F_{ij} of the form

F_{12} = c_{1} cosh φ + c_{2} sinh φ, F_{13} = F_{13}(x̃^{1}, r, x̃^{3}),
F_{14} = −c_{1} sinh φ − c_{2} cosh φ, F_{23} = c_{3} cosh φ + c_{4} sinh φ,
F_{24} = F_{24}(x̃^{1}, r, x̃^{3}), F_{34} = c_{3} sinh φ + c_{4} cosh φ,

(94) where the functions F_{13}(x̃^{1}, r, x̃^{3}), F_{24}(x̃^{1}, r, x̃^{3}), c_{i} = c_{i}(x̃^{1}, r, x̃^{3}) (i = 1, …, 4) satisfy the equations

\begin{matrix}\frac{∂c_{2}}{∂r} + \frac{c_{2}}{r} − \frac{λ}{r} \frac{∂c_{1}}{∂x̃^{3}} + \frac{∂F_{24}}{∂x̃^{1}} = 0, \frac{∂c_{1}}{∂x̃^{3}} − \frac{∂F_{13}}{∂r} + \frac{∂c_{3}}{∂x̃^{1}} = 0, \\ \frac{∂c_{4}}{∂r} + \frac{c_{4}}{r} − \frac{λ}{r} \frac{∂c_{3}}{∂x̃^{3}} − \frac{∂F_{24}}{∂x̃^{3}} = 0, \frac{λ}{r} \frac{∂F_{13}}{∂x̃^{3}} − \frac{∂c_{4}}{∂x̃^{1}} − \frac{∂c_{2}}{∂x̃^{3}} = 0 \end{matrix}  (95)

and the system

\begin{matrix}\frac{∂c_{1}}{∂r} + \frac{λ}{r} \frac{∂c_{1}}{∂x̃^{3}} − \frac{c_{2}}{r} = 0, \frac{∂c_{2}}{∂r} + \frac{λ}{r} \frac{∂c_{2}}{∂x̃^{3}} − \frac{c_{1}}{r} = 0, \frac{∂F_{13}}{∂r} + \frac{λ}{r} \frac{∂F_{13}}{∂x̃^{3}} = 0, \\ \frac{∂c_{3}}{∂r} + \frac{λ}{r} \frac{∂c_{3}}{∂x̃^{3}} − \frac{c_{4}}{r} = 0, \frac{∂c_{4}}{∂r} + \frac{λ}{r} \frac{∂c_{4}}{∂x̃^{3}} − \frac{c_{3}}{r} = 0, \frac{∂F_{24}}{∂r} + \frac{λ}{r} \frac{∂F_{24}}{∂x̃^{3}} = 0 \end{matrix}  (96)

The transformation of coordinates is defined by formulae (91).

The class P_{2,6} of potentials A_{i}, that are invariant with respect to the group G_{2,6}, consists of the following fields [23]

A_{1} = A_{1}(x̃^{1}, x̃^{3} − λ ln r), A_{2} = C_{1} cosh φ + C_{2} sinh φ,

A_{3} = A_{3}(x̃^{1}, x̃^{3} − λ ln r), A_{4} = −C_{1} sinh φ − C_{2} cosh φ, (97)

where

C_{1} = a_{1}(x̃^{1}, x̃^{3} − λ ln r) cosh ln r + a_{2}(x̃^{1}, x̃^{3} − λ ln r) sinh ln r,
C_{2} = a_{1}(x̃^{1}, x̃^{3} − λ ln r) sinh ln r + a_{2}(x̃^{1}, x̃^{3} − λ ln r) cosh ln r.

(98)

For the field A_{i} = (−Φ, 0, 0, 0), Φ = Φ(x̃^{1}, x̃^{3} − λ ln r) = Φ(t_{1}, t_{2}), we obtain

F_{12} = F_{14} = −\frac{λ}{r}Φ_{2}e^{−φ}, F_{13} = Φ_{2},

F_{23} = F_{24} = F_{34} = 0 (Φ_{2} = ∂Φ/∂t_{2}). (99)

Proposition 44. Let Φ_{ij} = ∂^{2}Φ/∂t_{i}∂t_{j}. If the partial derivatives Φ_{21} and Φ_{22} are linearly independent, then the Maxwell space defined by the tensor (99) admits the twodimensional group G_{S} = G_{2,6}.

6.2. Three-dimensional subgroups.

6.2.1. The class C_{3,5}. The algebra L_{3,5} = L{e_{24}, e_{1}, e_{3}} corresponds to the group G_{3,5} generated by pseudo-rotations in the plane Ox^{2}x^{4} and by translations along the vectors of the Euclidean plane Ox^{1}x^{3}. The Maxwell space of the class C_{3,5} is defined by the tensor F_{ij} of the form

F_{12} = a_{1}(r) cosh φ + \frac{b_{1}}{r} sinh φ, F_{13} = b_{3},

F_{14} = −a_{1}(r) sinh φ − \frac{b_{1}}{r} cosh φ, F_{23} = a_{2}(r) cosh φ + \frac{b_{2}}{r} sinh φ,

F_{24} = F_{24}(r) F_{34} = a_{2}(r) sinh φ + \frac{b_{2}}{r} cosh φ,

(100)

where a_{1}(r), a_{2}(r) � F_{24}(r) are arbitrary functions, b_{k} = const, and the transformation of coordinates is defined by formulae (31).

Proposition 45. If F_{24}(r) ≠ const, and the functions a_{1}(r), a_{2}(r), 1/r are linearly independent, then the Maxwell space defined by the tensor (100) admits the three-di-mensional group G_{S} = G_{3,5}.

6.2.2. The class C_{3,7}. The algebra L_{3,7} = L{e_{24} + λe_{3}, e_{1}, e_{2} − e_{4}} corresponds to the group G_{3,7} generated by hyperbolic helices and by translations along the vectors of an isotropic plane. C_{3,7} = C_{2,5} ∩ C_{2,6}. The Maxwell space of the class C_{3,7} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = \frac{λ}{r} e^{−φ} Φ (r e^{−x̃^{3}/λ}) , F_{13} = Φ (r e^{−x̃^{3}/λ}) ,

F_{24} = Ψ (r e^{−x̃^{3}/λ}) , F_{23} = c_{3}(r, x̃^{3}) cosh φ + c_{4}(r, x̃^{3}) sinh φ,

F_{34} = c_{3}(r, x̃^{3}) sinh φ + c_{4}(r, x̃^{3}) cosh φ,

(101)

where Φ(u) and Ψ(u) are arbitrary functions of one variable, the functions c_{3} = c_{3}(r, x̃^{3}) and c_{4} = c_{4}(r, x̃^{3}) are satisfied to equations

\begin{matrix}\frac{∂c_{4}}{∂r} + \frac{c_{4}}{r} − \frac{λ}{r} \frac{∂c_{3}}{∂x̃^{3}} − \frac{∂F_{24}}{∂x̃^{3}} = 0, \\ \frac{∂c_{3}}{∂r} + \frac{λ}{r} \frac{∂c_{3}}{∂x̃^{3}} − \frac{c_{4}}{r} = 0, \\ \frac{∂c_{4}}{∂r} + \frac{λ}{r} \frac{∂c_{4}}{∂x̃^{3}} − \frac{c_{3}}{r} = 0, \end{matrix}  (102)

and the transformation of coordinates is defined by formulae (91).

The class P_{3,7} of potentials A_{i} that are invariant with respect to the group G_{3,7} consists of the fields defined by (97) and (98), where the functions do not depend on x̃^{1}. For the field A_{i} = (−Φ, 0, 0, 0), Φ = Φ(x̃^{3} − λ ln r) = Φ(t), we obtain

F_{12} = F_{14} = −\frac{λ}{r}Φ′e^{−φ}, F_{13} = Φ′, F_{23} = F_{24} = F_{34} = 0 (Φ′ = dΦ/dt).  (103)

Proposition 46. If Φ′′ ≠ 0, then the Maxwell space defined by the tensor (103) admits the three-dimensional group G_{S} = G_{3,7}.

6.2.3. The class C_{3,16}. The algebra L_{3,16} = L{e_{12} − e_{14}, e_{24} + λe_{1} + μe_{3}, e_{2} − e_{4}} corresponds to the group G_{3,16} generated by parabolic rotations, hyperbolic helices, and by translations along an isotropic straight line. For description of the class C_{3,16} we use the change

x^{1} = λφ + x̃^{1}, x^{2} = r cosh φ, x^{3} = μφ + x̃^{3}, x^{4} = r sinh φ,  (104)

which contains, in particular, transformations (25), (31), and (91). The Maxwell space is defined by the tensor F_{ij} of the form

F_{12} = c_{1} cosh φ + c_{2} sinh φ, F_{14} = −c_{1} sinh φ − c_{2} cosh φ,

F_{13} = b_{1}x̃^{1} + b_{2}, F_{23} = c_{3} cosh φ + c_{4} sinh φ,
F_{34} = c_{3} sinh φ + c_{4} cosh φ, F_{24} = a_{2},

(105)

where

c_{1} = −c_{2} = \frac{a_{2}x̃^{1} + a_{3}}{r} , c_{3} = −\frac{1}{r} (\frac{b_{1}}{2} (x̃^{1})^{2} + b_{2}x̃^{1} + b_{3}) , c_{4} = b_{1}r − c_{3}, (106)

and the functions a_{k} = a_{k}(r, x̃^{3}) (k = 2, 3), b_{l} = b_{l}(r, x̃^{3}) (l = 1, 2, 3) are defined by the following formulae:

a) for the class C_{3,16a} that corresponds to the algebra L_{3,16a} = L{e_{12} −e_{14}, e_{24}, e_{2} −e_{4}} (L_{3,16} for λ = μ = 0),

\begin{matrix}a_{2} = Φ_{1}(x̃^{3}), a_{3} = Φ_{2}(x̃^{3}), b_{1} = Φ′_{1}(x̃^{3}), \\ b_{2} = Φ′_{2}(x̃^{3}), b_{3} = Φ_{3}(x̃^{3}) − \frac{r^{2}}{2} Φ′_{1}(x̃^{3}), \end{matrix}  (107)

where Φ_{1}(x̃^{3}), Φ_{2}(x̃^{3}), Φ_{3}(x̃^{3}) are arbitrary functions and the change is defined by (31);

b) for the class C_{3,16b} that corresponds to the algebra L{e_{12} − e_{14}, e_{24} + λe_{1}, e_{2} − e_{4}} (L_{3,16} for λ ≠ 0, μ = 0),

a_{2} = Φ_{1}(x̃^{3}), a_{3} = Φ_{2}(x̃^{3}) − λ ln r Φ_{1}(x̃^{3}),

b_{2} = Φ′_{2}(x̃^{3}) + (λ − λ ln r) Φ′_{1}(x̃^{3}), b_{1} = Φ′_{1}(x̃^{3}),

b_{3} = Φ_{3}(x̃^{3}) + (\frac{λ^{2}}{2} ln^{2} r − λ^{2} ln r − \frac{r^{2}}{2} ) Φ′_{1}(x̃^{3}) − λ ln r Φ′_{2}(x̃^{3}),

(108)

and the change is defined by (25) (for λ = 0 the formulae (108) are transformed in (107));

c) for the class C_{3,16c} that corresponds to the algebra L_{3,16c} = L_{3,16} (λ ≠ 0, μ ≠ 0),

a_{2} = Φ_{1}(u), a_{3} = Φ_{2}(u) − \frac{λv}{2μ}Φ_{1}(u), b_{2} = Φ_{4}(u) − \frac{λv}{2μ}Φ_{3}(u),

b_{1} = Φ_{3}(u), b_{3} = \frac{λ^{2}v^{2}}{8μ^{2}} Φ_{3}(u) − \frac{λv}{2μ}Φ_{4}(u) − \frac{r^{2}}{2} Φ_{3}(u) + Φ_{5}(u),

(109)

where

u = x̃^{3} − μ ln r, v = x̃^{3} + μ ln r,  (110)

Φ_{1}(u), Φ_{2}(u), Φ_{5}(u) are arbitrary functions,

\begin{matrix}Φ_{3}(u) = −\frac{1}{μ}e^{u/μ} ∫ Φ′_{1}(u)e^{−u/μ} du, \\ Φ_{4}(u) = −\frac{1}{μ}e^{u/μ} ∫ (Φ′_{2}(u) − \frac{λ}{2μ}Φ_{1}(u) + \frac{λ}{2}Φ_{3}(u)) e^{−u/μ} du, \end{matrix}  (111)

and the change is defined by (104);

d) for the class C_{3,16d} that corresponds to the algebra L{e_{12} − e_{14}, e_{24} + μe_{3}, e_{2} − e_{4}} (L_{3,16} for λ = 0, μ ≠ 0),

\begin{matrix}a_{2} = Φ_{1}(u), a_{3} = Φ_{2}(u), b_{1} = Φ_{3}(u), \\ b_{2} = Φ_{4}(u), b_{3} = −\frac{r^{2}}{2} Φ_{3}(u) + Φ_{5}(u), \end{matrix}  (112)

where u is defined in (110) and Φ_{1}(u), Φ_{2}(u), Φ_{5}(u) are arbitrary functions,

Φ_{3}(u) = −\frac{1}{μ}e^{u/μ} ∫ Φ′_{1}(u)e^{−u/μ} du, Φ_{4}(u) = −\frac{1}{μ}e^{u/μ} ∫ Φ′_{2}(u)e^{−u/μ} du,  (113)

and the change is defined by (91) (λ ↦ μ)

x^{1} = x̃^{1}, x^{2} = r cosh φ, x^{3} = μφ + x̃^{3}, x^{4} = r sinh φ.  (114)

For λ = 0 the formulae (109) and (111) are transformed in (112) and (113) .

For classes C_{3,16a} and C_{3,16b} we take Φ_{1} = Φ_{3} = 0 and Φ_{2} = Φ(x̃^{3}); we obtain the following examples of the tensors F_{ij}:

F_{12} = F_{14} = \frac{1}{r}Φ(x̃^{3})e^{−φ}, F_{13} = Φ′(x̃^{3}),

F_{24} = 0, F_{34} = −F_{23} = \frac{1}{r}x̃^{1}Φ′(x̃^{3})e^{−φ} (115)

and

\begin{matrix}F_{12} = F_{14} = \frac{1}{r}Φ(x̃^{3})e^{−φ}, F_{13} = Φ′(x̃^{3}), F_{24} = 0, \\ F_{34} = −F_{23} = \frac{1}{r}(x̃^{1} − λ ln r)Φ′(x̃^{3})e^{−φ}. \end{matrix}  (116)

Proposition 47. If Φ′′(x̃^{3}) ≠ 0, then the Maxwell space defined by the tensor (115) admits the three-dimensional group G_{S} = G_{3,16a}, and the Maxwell space defined by the tensor (116) admits the group G_{S} = G_{3,16b}.

For classes C_{3,16c} and C_{3,16d} we take Φ_{2} = Φ(u) = Φ(x̃^{3} − μ ln r) and Φ_{1} = Φ_{5} = 0. Then Φ_{3} = 0 and

Φ_{4}(u) = −\frac{1}{μ}e^{u/μ} ∫ Φ′(u)e^{−u/μ} du.  (117)

We obtain the following examples of the tensors F_{ij}:

F_{12} = F_{14} = \frac{1}{r}Φ(x̃^{3} − μ ln r)e^{−φ}, F_{13} = Φ_{4}(x̃^{3} − μ ln r), F_{24} = 0,

F_{34} = −F_{23} = \frac{1}{2μr}e^{−φ}Φ_{4}(x̃^{3} − μ ln r) · (2μx̃^{1} − λx̃^{3} − λμ ln r)

(118)

and

F_{12} = F_{14} = \frac{1}{r}Φ(x̃^{3} − μ ln r)e^{−φ}, F_{13} = Φ_{4}(x̃^{3} − μ ln r),

F_{34} = −F_{23} = \frac{x̃^{1}}{2r}e^{−φ}Φ_{4}(x̃^{3} − μ ln r), F_{24} = 0.

(119)

Proposition 48. If Φ′′(u) ≠ 0, then the Maxwell space defined by the tensor (118) admits the three-dimensional group G_{S} = G_{3,16c} and the Maxwell space defined by the tensor (119) admits the group G_{S} = G_{3,16d}.

6.2.4. The class C_{3,21}. The algebra L_{3,21} = L{e_{12}, e_{14}, e_{24}} corresponds to the group G_{3,21} generated by rotations in the plane Ox^{1}x^{2} and by pseudo-rotations in planes Ox^{1}x^{4} and Ox^{2}x^{4}. The Maxwell space of the class C_{3,21} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{24} = 0, F_{13} = x^{1} Φ(x^{3}), F_{23} = x^{2} Φ(x^{3}), F_{34} = x^{4} Φ(x^{3}), (120)

where Φ(x^{3}) is an arbitrary function.

Proposition 49. If Φ(x^{3}) ≠ const, then the Maxwell space defined by the tensor (120) admits the three-dimensional group G_{S} = G_{3,21}.

6.3. Four-dimensional subgroups.

6.3.1. The class C_{4,5}. The algebra L_{4,5} = L{e_{24}, e_{1}, e_{3}, e_{2} + e_{4}} corresponds to the group G_{4,5} generated by pseudo-rotations in the plane Ox^{2}x^{4} and by translations along the vectors of the isotropic hyperplane. Since L_{3,5} ⊂ L_{4,5}, then C_{4,5} ⊂ C_{3,5}. The Maxwell space of the class C_{4,5} is defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} = −F_{14} = \frac{b_{1}}{x^{2} − x^{4}} , F_{23} = F_{34} = \frac{b_{2}}{x^{2} − x^{4}} , \\ F_{13} = b_{3}, F_{24} = b_{4} (b_{k} = const). \end{matrix}  (121)

Proposition 50. Suppose one of the following conditions is satisfied: 1) b_{1} ≠ 0 and b_{3} ≠ 0, 2) b_{1} ≠ 0 and b_{4} ≠ 0, 3) b_{2} ≠ 0 and b_{3} ≠ 0, 4) b_{2} ≠ 0 and b_{4} ≠ 0; then the Maxwell space defined by the tensor (121) admits the four-dimensional group G_{S} = G_{4,5}.

6.3.2. The class C_{4,13}. The algebra L_{4,13} = L{e_{12} − e_{14}, e_{24} + λe_{1}, e_{3}, e_{2} − e_{4}} corresponds to the group G_{4,13} generated by parabolic rotations, by hyperbolic helices, and by translations along the vectors of an isotropic plane. Since L_{3,16b} ⊂ L_{4,13}, then C_{4,13} ⊂ C_{3,16b}. The Maxwell space of the class C_{4,13} is defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} = F_{14} = \frac{1}{r} (K_{1}x̃^{1} − K_{1}λ ln r + K_{2}) e^{−φ}, \\ F_{23} = −F_{34} = \frac{K_{3}}{r} e^{−φ}, F_{13} = 0, F_{24} = K_{1}, \end{matrix}  (122)

where K_{1}, K_{2}, K_{3} are arbitrary constants, and the transformation of coordinates is defined by formulae (25).

Proposition 51. If K_{1} ≠ 0, then the Maxwell space defined by the tensor (122) admits the four-dimensional group G_{S} = G_{4,13}.

6.3.3. The class C_{4,14}. The algebra L_{4,14} = L{e_{12} − e_{14}, e_{24} + λe_{3}, e_{1} + νe_{3}, e_{2} − e_{4}} corresponds to the group G_{4,14} generated by parabolic rotations, by hyperbolic helices, and by translations along the vectors of an isotropic plane. The group G_{4,14} is not conjugate to G_{4,13}. Since L_{3,16d} ⊂ L_{4,14}, then C_{4,14} ⊂ C_{3,16b}. The Maxwell space of the class C_{4,14} is defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} = F_{14} = \frac{1}{r} (K_{1}x̃^{1} − \frac{K_{1}}{ν} (x̃^{3} − λ ln r) + K_{2}) e^{−φ}, F_{13} = −\frac{K_{2}}{ν} , \\ F_{23} = −F_{34} = \frac{1}{r} (\frac{K_{1}}{ν} x̃^{1} − \frac{K_{1}}{ν^{2}} (x̃^{3} − λ ln r) + K_{3}) e^{−φ}, F_{24} = K_{1}, \end{matrix}  (123)

where K_{i} = const and the transformation of coordinates is defined by formulae (91).

Proposition 52. If K_{1} ≠ 0, then the Maxwell space defined by the tensor (123) admits the four-dimensional group G_{S} = G_{4,14}.

For ν = 0, i. e., for the algebra L_{4,14a} = L{e_{12} − e_{14}, e_{24} + λe_{3}, e_{1}, e_{2} − e_{4}}, the class C_{4,14a} is defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} = F_{14} = \frac{K_{1}}{r} e^{−φ}, F_{13} = F_{24} = 0, \\ F_{23} = −F_{34} = −\frac{1}{r}Ψ(x̃^{3} − λ ln r)e^{−φ}, \end{matrix}  (124)

where Ψ(u) is an arbitrary function.

Proposition 53. If Ψ′(u) ≠ 0, then the Maxwell space defined by the tensor (124) admits the four-dimensional group G_{S} = G_{4,14a}.

6.3.4. The class C_{4,15}. The algebra L_{4,15} = L{e_{12} − e_{14}, e_{23} + e_{34}, e_{24} + λe_{1}, e_{2} − e_{4}} corresponds to the group G_{4,15} generated by two one-dimensional subgroups of parabolic rotations, by hyperbolic helices, and by translations along an isotropic straight line. Since L_{3,16b} ⊂ L_{4,15}, then C_{4,15} ⊂ C_{3,16b}. The Maxwell space of the class C_{4,15} is defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} = F_{14} = \frac{1}{r}e^{−φ}(K_{1}x̃^{1} − λK_{1} ln r + K_{2}), F_{13} = 0, \\ F_{23} = −F_{34} = −\frac{1}{r}e^{−φ}(K_{1}x̃^{3} + K_{3}), F_{24} = K_{1}, \end{matrix}  (125)

where K_{i} (i = 1, 2, 3) are arbitrary constants, and the transformation of coordinates is defined by formulae (25).

Proposition 54. If K_{1} ≠ 0, then the Maxwell space defined by the tensor (125) admits the four-dimensional group G_{S} = G_{4,15}.

6.3.5. The class C_{4,19}. The algebra L_{4,19} = L{e_{12}, e_{14}, e_{24}, e_{3}} corresponds to the group G_{4,19} generated by rotations, by pseudo-rotations, and by translations along a space-like straight line. Since L_{3,21} ⊂ L_{4,19}, then C_{4,19} ⊂ C_{3,21}. The Maxwell space of the class C_{4,19} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{24} = 0, F_{13} = Kx^{1},

F_{23} = Kx^{2}, F_{34} = Kx^{4} (K = const). (126)

Proposition 55. If K ≠ 0, then the Maxwell space defined by the tensor (126) admits the four-dimensional group G_{S} = G_{4,19}.

6.4. Five- and six-dimensional subgroups.

6.4.1. For each algebra

L_{5,7} = L{e_{12} − e_{14}, e_{24}, e_{1}, e_{3}, e_{2} − e_{4}},
L_{5,8} = L{e_{12} − e_{14}, e_{23} + e_{34}, e_{24} + λe_{3}, e_{1}, e_{2} − e_{4}},

and

L_{6,6} = L{e_{12} − e_{14}, e_{23} + e_{34}, e_{24}, e_{1}, e_{3}, e_{2} − e_{4}}

there is the class C_{6,6} of Maxwell’s spaces defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} = F_{14} = \frac{K_{1}}{x^{2} + x^{4}} , F_{13} = F_{24} = 0, \\ F_{23} = −F_{34} = \frac{K_{2}}{x^{2} + x^{4}} , (K_{1}, K_{2} = const). \end{matrix}  (127)

Proposition 56. If K_{1} ≠ 0 or K_{2} ≠ 0, then the Maxwell space defined by the tensor (127) admits the six-dimensional group G_{S} = G_{6,6}.

6.4.2. For the algebra L_{6,9} = L{e_{12}, e_{14}, e_{24}, e_{1}, e_{2}, e_{4}} the corresponding class of Maxwell’s spaces is empty.

7. The Maxwell spaces that admit parabolic helices

In this section, we describe classes of the Maxwell spaces that admit parabolic helices and parabolic rotations and were not discussed in the previous sections. See details in [21].

7.1. Two-dimensional subgroups.

7.1.1. The classes C_{2,7a} and C_{2,7b}. The algebra L_{2,7} = L{e_{12} − e_{14} + λe_{2} + μe_{3}, e_{2} − e_{4}} corresponds to the group G_{2,7} generated by parabolic helices and by translations along an isotropic straight line. The algebra L_{2,7} is an extension of the algebra L_{1,4} by the vector ξ = e_{2} − e_{4}. Therefore, the classes C_{2,7a}, C_{2,7b}, and C_{2,7c} that corresponding to L_{2,7} for various values of λ and μ are obtained as restrictions of classes the C_{1,4a}, C_{1,4b}, and C_{1,4c} by using condition (2) for the vector ξ = e_{2} − e_{4}: ∂_{2}F_{ij} − ∂_{4}F_{ij} = 0. The Maxwell space of the class C_{2,7b} (λ = 0, μ ≠ 0) is defined by the tensor F_{ij} of the form (36), where C_{k} = C_{k}(x̃^{1}, x̃^{3}) are the smooth functions that satisfy (37) and satisfy the equations

\begin{matrix}\frac{μ}{x̃^{1}} \frac{∂C_{1}}{∂x̃^{3}} + \frac{∂C_{5}}{∂x̃^{3}} = 0, \frac{∂C_{1}}{∂x̃^{1}} + \frac{∂C_{6}}{∂x̃^{3}} = 0, \\ \frac{μ}{x̃^{1}} \frac{∂C_{3}}{∂x̃^{3}} − \frac{C_{2}}{x̃^{1}} − \frac{∂C_{2}}{∂x̃^{1}} + \frac{∂C_{7}}{∂x̃^{3}} = 0, \frac{μ}{x̃^{1}} \frac{∂C_{6}}{∂x̃^{3}} − \frac{C_{5}}{x̃^{1}} − \frac{∂C_{5}}{∂x̃^{1}} \end{matrix}  (128)

(the transformation of coordinates is defined by formulae (39)).

For μ = 0 the system (128) is simplified and integrated partially. We obtain the result. The Maxwell space of the class C_{2,7a} (λ = 0, μ = 0) is defined by the tensor F_{ij} of the form (36), where C_{k} = C_{k}(x̃^{1}, x̃^{3}) (k ≠ 5) are the smooth functions satisfying (37) and the equations

\frac{∂C_{1}}{∂x̃^{1}} + \frac{∂C_{6}}{∂x̃^{3}} = 0, \frac{C_{2}}{x̃^{1}} + \frac{∂C_{2}}{∂x̃^{1}} − \frac{∂C_{7}}{∂x̃^{3}} = 0,  (129)

and C_{5} = A/x̃^{1} (A = const, the transformation of coordinates is defined by formulae (35)).

Proposition 57. Suppose the functions Φ(t_{1}, t_{2}) = Φ (x^{2} + x^{4}, x^{3}), Φ_{1}, Φ_{2}, Φ_{12} and Φ_{22} are linearly independent (Φ_{k} = ∂Φ/∂t_{k}, Φ_{kl} = ∂^{2}Φ/∂t_{k}∂t_{l}); then the Maxwell space defined by the tensor

F_{12} = F_{14} = −Φ_{1} − \frac{Φ}{x^{2} + x^{4}} ,  F_{13} = −Φ_{2},
F_{23} = −F_{34} = \frac{x^{1}}{x^{2} + x^{4}} Φ_{2},  F_{24} = 0

admits the two-dimensional group G_{S} = G_{2,7a}.

In particular, this condition holds for the function Φ(t_{1}, t_{2}) = (t_{1}^{2} + t_{2}^{3})t_{2}.

Proposition 58. Let

Φ(t_{1}, t_{2}) = Φ (x^{2} + x^{4}, x^{3} + \frac{μx^{1}}{x^{2} + x^{4}} ) ,

Φ_{k} = ∂Φ/∂t_{k}, Φ_{kl} = ∂^{2}Φ/∂t_{k}∂t_{l}.

Suppose the partials Φ_{2}, Φ_{12}, and Φ_{22} are linearly independent; then the Maxwell space defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{24} = 0, F_{13} = \frac{μ}{x^{2} + x^{4}} Φ_{2},

F_{23} = −F_{34} = Φ_{1} − \frac{μx^{1}}{(x^{2} + x^{4})^{2}} Φ_{2},

admits the two-dimensional group G_{S} = G_{2,7b}.

In particular, this condition holds for the function Φ(t_{1}, t_{2}) = (t_{1}^{2} + t_{2})t_{2}.

7.1.2. The class C_{2,7c}. For λ ≠ 0 and μ = 0 we have the following result. The Maxwell space of the class C_{2,7c} is defined by the tensor F_{ij} of the form (36), where the functions C_{k} = C_{k}(x̃^{1}, x̃^{3}) (k ≠ 6) satisfy (37) and also satisfy the equations

\frac{C_{1}}{λ} − 2λ\frac{∂C_{3}}{∂x̃^{1}} − \frac{∂C_{7}}{∂x̃^{3}} = 0, 2λ\frac{∂C_{1}}{∂x̃^{1}} + \frac{∂C_{5}}{∂x̃^{3}} = 0,  (130)

and C_{6} = const (the coordinates {x̃^{i}} are related with {x^{i}} by means of formulae (41)).

Proposition 59. Let

Φ(t_{1}, t_{2}) = Φ (2λx^{1} + (x^{2} + x^{4})^{2} , x^{3}) ,
Φ_{k} = ∂Φ/∂t_{k}, Φ_{kl} = ∂^{2}Φ/∂t_{k}∂t_{l}.

Suppose the partials Φ_{1}, Φ_{11}, and Φ_{12} are linearly independent; then the Maxwell space defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{24} = 0, F_{13} = 2λΦ_{1}, F_{23} = −F_{34} = 2(x^{2} + x^{4})Φ_{1},

admits the two-dimensional group G_{S} = G_{2,7c}.

In particular, this condition holds for the function Φ(t_{1}, t_{2}) = t_{1}(t_{1}^{2} + t_{2}^{2}).

7.1.3. The class C_{2,8}. The algebra L_{2,8} = L{e_{12}−e_{14}+λe_{2}, e_{3}} corresponds to the group G_{2,8} generated by parabolic helices and by translations along a space-like straight line. Since L_{1,4c} ⊂ L_{2,8}, then C_{2,8} ⊂ C_{1,4c}. The Maxwell space of the class C_{2,8} is defined by the tensor F_{ij} of the form (36), where the functions C_{k} = C_{k}(x̃^{1}, x̃^{4}) satisfy (37) and the following equations,

\begin{matrix}\frac{C_{1}}{λ} + \frac{x̃^{1}}{2λ} \frac{∂C_{2}}{∂x̃^{4}} − 2λ\frac{∂C_{3}}{∂x̃^{1}} = 0, \frac{x̃^{1}}{2λ} \frac{∂C_{1}}{∂x̃^{4}} − λ\frac{∂C_{3}}{∂x̃^{4}} = 0, \\ 2\frac{∂C_{1}}{∂x̃^{1}} − \frac{∂C_{2}}{∂x̃^{4}} = 0, \frac{x̃^{1}}{2λ} \frac{∂C_{5}}{∂x̃^{4}} − 2λ\frac{∂C_{6}}{∂x̃^{1}} − λ\frac{∂C_{7}}{∂x̃^{4}} = 0, \end{matrix}  (131)

and the transformation of the coordinates is defined by formulae (41).

Proposition 60. The Maxwell space defined by the tensor F_{ij} is of the form

F_{12} = F_{14} = F_{24} = 0, F_{23} = 2(x^{2} + x^{4})Φ_{1} + (x^{1} + \frac{(x^{2} + x^{4})^{2}}{λ} ) Φ_{2},

F_{13} = 2λΦ_{1} + (x^{2} + x^{4})Φ_{2},

F_{34} = −2(x^{2} + x^{4})Φ_{1} − (λ + x^{1} + \frac{(x^{2} + x^{4})^{2}}{λ} ) Φ_{2},

where

Φ(t_{1}, t_{2}) =

Φ (2λx^{1} + (x^{2} + x^{4})^{2} , λx^{4} + x^{1}(x^{2} + x^{4}) + \frac{1}{3λ} (x^{2} + x^{4})^{3}) ,

admits the two-dimensional group G_{S} = G_{2,8} whenever the partials Φ_{k} = ∂Φ/∂t_{k} and Φ_{kl} = ∂^{2}Φ/∂t_{k}∂t_{l} are linearly independent.

In particular, this condition holds for the function Φ(t_{1}, t_{2}) = t_{1}(t_{1}^{2} + t_{2}^{2}).

7.1.4. The class C_{2,11}. The algebra L_{2,11} = L{e_{12} −e_{14} +λe_{1} +μe_{3}, e_{23}+ e_{34} −μe_{1} +λe_{3}} (λ = 0, μ ≠ 0 ∼ λ ≠ 0, μ = 0) corresponds to the group G_{2,11} generated by two one-di-mensional subgroups of parabolic helices. The Maxwell space of the class C_{2,11} is defined by the tensor F_{ij} of the form (36), where the functions C_{k} = C_{k}(x̃^{1}, x̃^{3}, x̃^{4}) satisfy (37), (38b), (38e), (38g), (40), and the equations

\frac{(x̃^{1})^{2} − μ^{2}}{x̃^{1}} \frac{∂C_{5}}{∂x̃^{3}} − x̃^{1}x̃^{3} \frac{∂C_{5}}{∂x̃^{4}} = 0,  (132a)
\frac{(x̃^{1})^{2} − μ^{2}}{x̃^{1}} \frac{∂C_{6}}{∂x̃^{3}} − x̃^{1}x̃^{3} \frac{∂C_{6}}{∂x̃^{4}} − C_{1} + \frac{μ}{x̃^{1}} C_{5} = 0,  (132b)
\frac{(x̃^{1})^{2} − μ^{2}}{x̃^{1}} \frac{∂C_{7}}{∂x̃^{3}} − x̃^{1}x̃^{3} \frac{∂C_{7}}{∂x̃^{4}} + C_{2} − \frac{μ}{x̃^{1}} C_{6} = 0,  (132c)
\frac{(x̃^{1})^{2} − μ^{2}}{x̃^{1}} \frac{∂C_{1}}{∂x̃^{3}} − x̃^{1}x̃^{3} \frac{∂C_{1}}{∂x̃^{4}} = 0,  (132d)
\frac{(x̃^{1})^{2} − μ^{2}}{x̃^{1}} \frac{∂C_{2}}{∂x̃^{3}} − x̃^{1}x̃^{3} \frac{∂C_{2}}{∂x̃^{4}} + C_{5} + \frac{μ}{x̃^{1}} C_{1} = 0,  (132e)
\frac{(x̃^{1})^{2} − μ^{2}}{x̃^{1}} \frac{∂C_{3}}{∂x̃^{3}} − x̃^{1}x̃^{3} \frac{∂C_{3}}{∂x̃^{4}} + C_{6} + \frac{μ}{x̃^{1}} C_{2} = 0  (132f)

(the transformation of coordinates is defined by formulae (39)).

Proposition 61. The Maxwell space defined by the tensor F_{ij} of the form

F_{12} = \frac{1}{μ^{2} − (x^{2} + x^{4})^{2}} \{μ^{3} (μx^{2} − x^{1}x^{3}) + (x^{2} + x^{4}) [μ^{4} − μ^{2} (x^{1})^{2}] +

+ (x^{2} + x^{4})^{2} (4μ x^{1}x^{3} − 5μ^{2}x^{2}) +

(x^{2} + x^{4})^{3} [\frac{5}{2} (x^{1})^{2} + \frac{3}{2} (x^{3})^{2} − 2μ^{2}] +

+4x^{2} (x^{2} + x^{4})^{4} + (x^{2} + x^{4})^{5}} ,

F_{13} = − (x^{2} + x^{4}) (μ x^{1} + x^{3} (x^{2} + x^{4})) ,

F_{14} = F_{12} − (x^{2} + x^{4}) [μ^{2} − (x^{2} + x^{4})^{2}] ,

F_{24} = − (x^{2} + x^{4}) (μ x^{3} + x^{1} (x^{2} + x^{4})) ,

F_{34} = −F_{23} = \frac{1}{μ^{2} − (x^{2} + x^{4})^{2}} ·

\{μ^{3}x^{2} (x^{2} + x^{4}) + \frac{μ}{2} [(x^{1})^{2} + (x^{3})^{2}] (x^{2} + x^{4})^{2} +

+ (x^{1}x^{3} − μ x^{2}) (x^{2} + x^{4})^{3}} .

admits the two-dimensional group G_{S} = G_{2,11}.

7.1.5. The class C_{2,11a}. The algebra L_{2,11a} = L{e_{12}−e_{14}, e_{23}+e_{34}} (L_{2,11} for λ = μ = 0) corresponds to the group G_{2,11a} generated by two one-dimensional subgroups of parabolic rotations. The Maxwell space of the class C_{2,11a} is defined by the tensor F_{ij} of the form (36), where the functions C_{k} = C_{k}(x̃^{1}, x̃^{3}, x̃^{4}) satisfy (37), (38), and the following equations

\begin{matrix}\frac{∂C_{5}}{∂x̃^{3}} − x̃^{3} \frac{∂C_{5}}{∂x̃^{4}} = 0, x̃^{1} (\frac{∂C_{6}}{∂x̃^{3}} − x̃^{3} \frac{∂C_{6}}{∂x̃^{4}} ) − C_{1} = 0, \\ x̃^{1} (\frac{∂C_{7}}{∂x̃^{3}} − x̃^{3} \frac{∂C_{7}}{∂x̃^{4}} ) + C_{2} = 0, \frac{∂C_{1}}{∂x̃^{3}} − x̃^{3} \frac{∂C_{1}}{∂x̃^{4}} = 0, \\ x̃^{1} (\frac{∂C_{2}}{∂x̃^{3}} − x̃^{3} \frac{∂C_{2}}{∂x̃^{4}} ) + C_{5} = 0, \\ x̃^{1} (\frac{∂C_{3}}{∂x̃^{3}} − x̃^{3} \frac{∂C_{3}}{∂x̃^{4}} ) + C_{6} = 0 \end{matrix}  (133)

(the transformation of coordinates is defined by formulae (35)).

Proposition 62. Let

Φ = Φ(t_{1}, t_{2}) = Φ (x^{2} + x^{4}, \frac{1}{2} ((x^{1})^{2} + (x^{3})^{2}) + x^{2}(x^{2} + x^{4})) ,

Φ_{k} = ∂Φ/∂t_{k}, Φ_{kl} = ∂^{2}Φ/∂t_{k}∂t_{l}. Suppose the functions Φ, Φ_{1}, Φ_{2}, and Φ_{22} are linearly independent; then the Maxwell space defined by the tensor F_{ij} of the form

F_{12} = \frac{1}{x^{2} + x^{4}} Φ + Φ_{1} + (\frac{(x^{1})^{2}}{x^{2} + x^{4}} + 2x^{2} + x^{4}) Φ_{2},

F_{13} = x^{3}Φ_{2}, F_{24} = x^{1}Φ_{2},

F_{14} = \frac{1}{x^{2} + x^{4}} Φ + Φ_{1} + (\frac{(x^{1})^{2}}{x^{2} + x^{4}} + x^{2}) Φ_{2},

F_{23} = −F_{34} = −\frac{x^{1}x^{3}}{x^{2} + x^{4}} Φ_{2}

admits the two-dimensional group G_{S} = G_{2,11a}.

7.1.6. The class C_{2,12}. The algebra L_{2,12} = L{e_{12} − e_{14}, e_{24} + λe_{3}} corresponds to the group G_{2,12} generated by parabolic rotations and by hyperbolic helices. The Maxwell space of the class C_{2,12} is defined by the tensor F_{ij} of the form (36), where the functions C_{k} = C_{k}(x̃^{1}, x̃^{3}, x̃^{4}) satisfy (37), (38) and to the following equations

x̃^{1} \frac{∂C_{1}}{∂x̃^{1}} + λ\frac{∂C_{1}}{∂x̃^{3}} + (x̃^{1})^{2} \frac{∂C_{1}}{∂x̃^{4}} − C_{1} = 0,  (134a)
x̃^{1} \frac{∂C_{2}}{∂x̃^{1}} + λ\frac{∂C_{2}}{∂x̃^{3}} + (x̃^{1})^{2} \frac{∂C_{2}}{∂x̃^{4}} = 0,  (134b)
x̃^{1} \frac{∂C_{3}}{∂x̃^{1}} + λ\frac{∂C_{3}}{∂x̃^{3}} + (x̃^{1})^{2} \frac{∂C_{3}}{∂x̃^{4}} − C_{4} = 0,  (134c)
x̃^{1} \frac{∂C_{5}}{∂x̃^{1}} + λ\frac{∂C_{5}}{∂x̃^{3}} + (x̃^{1})^{2} \frac{∂C_{5}}{∂x̃^{4}} − C_{5} = 0,  (134d)
x̃^{1} \frac{∂C_{6}}{∂x̃^{1}} + λ\frac{∂C_{6}}{∂x̃^{3}} + (x̃^{1})^{2} \frac{∂C_{6}}{∂x̃^{4}} = 0,  (134e)
x̃^{1} \frac{∂C_{7}}{∂x̃^{1}} + λ\frac{∂C_{7}}{∂x̃^{3}} + (x̃^{1})^{2} \frac{∂C_{7}}{∂x̃^{4}} + C_{8} = 0.  (134f)

(the transformation of coordinates is defined by formulae (35)).

7.1.7. The class C_{2,12a}. The algebra L_{2,12a} = L{e_{12} − e_{14}, e_{24}} (L_{2,12} for λ = 0) corresponds to the group G_{2,12a}, generated by parabolic rotations and by pseudo-rotations The Maxwell space of the class C_{2,12a} is defined by the tensor F_{ij} of the form (36), where functions

C_{1} = x̃^{1}Φ_{1} (x̃^{3}, x̃^{4} − \frac{1}{2} (x̃^{1})^{2}) , C_{2} = Φ_{2} (x̃^{3}, x̃^{4} − \frac{1}{2} (x̃^{1})^{2}) ,

C_{3} = −\frac{x̃^{1}}{2} Φ_{1} (x̃^{3}, x̃^{4} − \frac{1}{2} (x̃^{1})^{2}) + \frac{1}{x̃^{1}} Φ_{3} (x̃^{3}, x̃^{4} − \frac{1}{2} (x̃^{1})^{2}) ,

C_{5} = x̃^{1}Φ_{4} (x̃^{3}, x̃^{4} − \frac{1}{2} (x̃^{1})^{2}) , C_{6} = Φ_{5} (x̃^{3}, x̃^{4} − \frac{1}{2} (x̃^{1})^{2}) ,

C_{7} = −\frac{x̃^{1}}{2} Φ_{4} (x̃^{3}, x̃^{4} − \frac{1}{2} (x̃^{1})^{2}) + \frac{1}{x̃^{1}} Φ_{6} (x̃^{3}, x̃^{4} − \frac{1}{2} (x̃^{1})^{2}) ,

C_{4} = −C_{1} − C_{3}, C_{8} = C_{5} + C_{7} (135) satisfy (38) (the transformation of coordinates is defined by formulae (35)).

The potential of the form

A_{i} = (Φ, −\frac{x^{1}}{x^{2} + x^{4}} Φ, 0, −\frac{x^{1}}{x^{2} + x^{4}} Φ) ,
Φ = Φ(t_{1}, t_{2}) = Φ(x^{3}, (x^{1})^{2} + (x^{2})^{2} − (x^{4})^{2})

belongs to the class P_{2,12a}; it generates the following tensor F_{ij}:

\begin{matrix}F_{12} = −\frac{1}{x^{2} + x^{4}} Φ − (\frac{2(x^{1})^{2}}{x^{2} + x^{4}} + 2(x^{2})^{2}) Φ_{2}, \\ F_{14} = −\frac{1}{x^{2} + x^{4}} Φ − (\frac{2(x^{1})^{2}}{x^{2} + x^{4}} − 2(x^{4})^{2}) Φ_{2}, \\ F_{13} = −Φ_{1}, F_{24} = −2x^{1}Φ_{2}, F_{23} = −F_{34} = \frac{x^{1}}{x^{2} + x^{4}} Φ_{1} \end{matrix}  (136)

(Φ_{k} = ∂Φ/∂t_{k}, Φ_{kl} = ∂^{2}Φ/∂t_{k}∂t_{l}).

Proposition 63. The Maxwell space defined by the tensor F_{ij} of the form (136) admits the two-dimensional group G_{S} = G_{2,12a} whenever the functions x^{1}Φ_{11} −2x^{1}x^{3}Φ_{12}, F_{12} + x^{2}Φ_{11} − 2x^{2}x^{3}Φ_{12}, and 2x^{3}x^{4}Φ_{12} − x^{4}Φ_{11} are linearly independent.

In particular, this condition holds for the function Φ = t_{1}t_{2}.

7.2. Three-dimensional subgroups.

7.2.1. The class C_{3,8}. The algebra L_{3,8} = L{e_{12} − e_{14} + λe_{2}, e_{3}, e_{2} − e_{4}} corresponds to the group G_{3,8} generated by parabolic helices and by translations along the vectors of the isotropic plane. The Maxwell space of the class C_{3,8} is defined by the tensor F_{ij} of the form (36), where

C_{1} = a_{1}, C_{3} = \frac{a_{1}}{2λ^{2}} x̃^{1} + a_{2}, C_{6} = a_{3} (a_{i} = const),  (137)

and the functions C_{k} = C_{k}(x̃^{1}) (k = 2, 4, 5, 7, 8) satisfy (37) (the coordinates {x̃^{i}} and {x^{i}} are related by (41)).

Let a_{1} = A, a_{2} = a_{3} = 0, C_{k} = 0 (k = 2, 4, 5, 7, 8); taking into account the change (41), we obtain

\begin{matrix}F_{12} = F_{14} = F_{24} = 0, F_{13} = \frac{A}{λ}(x^{2} + x^{4}), \\ F_{23} = \frac{A}{λ^{2}} (λ^{2} + λx^{1} + (x^{2} + x^{4})^{2}), F_{34} = −A − F_{23}. \end{matrix}  (138)

Proposition 64. If A ≠ 0, then the Maxwell space defined by the tensor F_{ij} of form (138) admits the three-dimensional group G_{S} = G_{3,8}.

7.2.2. The class C_{3,9}. The algebra L_{3,9} = L{e_{12}−e_{14}+λe_{2}+μe_{3}, e_{1}, e_{2}−e_{4}} corresponds to the group G_{3,9} generated by parabolic rotations or parabolic helices with various axes and by translations along the vectors of an isotropic plane. We consider three cases.

a) λ = μ = 0. The Maxwell space of the class C_{3,9a} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = Φ(x^{2} + x^{4}), F_{13} = F_{24} = 0, F_{34} = −F_{23} = Ψ(x^{2} + x^{4}, x^{3}).  (139)

where Φ(u) � Ψ(u, v) are arbitrary functions.

Proposition 65. If Φ(u) ≠ const and ∂Ψ/∂x^{3} ≠ 0, then the Maxwell space defined by the tensor F_{ij} of the form (139) admits the three-dimensional group G_{S} = G_{3,9a}.

b) λ = 0, μ ≠ 0. The Maxwell space of the class C_{3,9b} is defined by the tensor F_{ij} of the form (36), where

\begin{matrix}_{1} = −\frac{a_{1}}{μ} x̃^{1} + a_{2}, C_{2} = \frac{x̃^{3}}{μ} (−\frac{a_{1}}{μ} x̃^{1} + a_{2}) + μΦ(x̃^{1}), \\ C_{3} = \frac{(x̃^{3})^{2}}{2μ^{2}} (−\frac{a_{1}}{μ} x̃^{1} + a_{2}) + x̃^{3}Φ(x̃^{1}) + Ψ_{1}(x̃^{1}), \\ C_{4} = −C_{1} − C_{3}, C_{5} = a_{1}, C_{6} = \frac{a_{1}}{μ} x̃^{3} − μ^{2}Φ′(x̃^{1}), \\ C_{7} = −\frac{a_{1}}{2μ^{2}} (x̃^{3})^{2} + μx̃^{3}Φ′(x̃^{1}) + Ψ_{2}(x̃^{1}), C_{8} = C_{5} + C_{7}; \end{matrix}  (140)

a_{1}, a_{2} = const, and Φ(x̃^{1}), Ψ_{1}(x̃^{1}), Ψ_{2}(x̃^{1}) are arbitrary functions (the coordinates {x̃^{i}} and {x^{i}} are related by (39)).

Set a_{1} = a_{2} = 0 in (140) and let Ψ_{1}(x̃^{1}) = Ψ_{2}(x̃^{1}) = 0; taking into account change (39), we obtain

\begin{matrix}F_{12} = F_{14} = μx^{3}Φ′(x^{2} + x^{4}), F_{13} = μΦ(x^{2} + x^{4}), \\ F_{23} = −F_{34} = x^{3}Φ(x^{2} + x^{4}), F_{24} = −μ^{2}Φ′(x^{2} + x^{4}). \end{matrix}  (141)

Proposition 66. If the functions Φ(x̃^{1}) and Φ′(x̃^{1}) are linearly independent, then the Maxwell space defined by the tensor F_{ij} of the form (140) admits the three-dimensional group G_{S} = G_{3,9b}.

c) λ ≠ 0, μ = 0. The Maxwell space of the class C_{3,9c} is defined by the tensor F_{ij} of the form (36), where C_{2} = C_{2}(x̃^{3}), C_{3} = C_{3}(x̃^{3}), and C_{7} = C_{7}(x̃^{3}) are arbitrary functions and

\begin{matrix}C_{1} = λC′_{7}(x̃^{3}), C_{4} = −λC′_{7}(x̃^{3}) − C_{3}(x̃^{3}), C_{5} = A, \\ C_{6} = B, C_{8} = C_{7}(x̃^{3}) + A (A, B = const) \end{matrix}  (142)

(the transformation of coordinates is defined by formulae (41) and the prime denotes differentiation with respect to x̃^{3}.)

Let C_{2} = Φ(x̃^{3}), C_{3} = C_{7} = A = B = 0; taking into account the change (41), we obtain

F_{12} = F_{14} = F_{24} = 0, F_{13} = Φ(x^{3}), F_{23} = −F_{34} = \frac{x^{2} + x^{4}}{λ} Φ(x^{3}).  (143)

Proposition 67. If Φ′(x^{3}) ≠ 0, then the Maxwell space defined by the tensor F_{ij} of the form (142) admits the three-dimensional group G_{S} = G_{3,9c}.

7.2.3. The class C_{3,10}. The algebra L_{3,10} = L{e_{12} − e_{14} + λe_{2}, e_{1} + μe_{3}, e_{2} − e_{4}} (λ ≠ 0, μ ≠ 0) is an extension of the algebra L_{2,7c} = L{e_{12} −e_{14} +λe_{2}, e_{2} −e_{4}} that corresponds to the class C_{2,7c}; hence C_{3,10} ⊂ C_{2,7c}. The Maxwell space of the class C_{3,10} is defined by the tensor F_{ij} of the form (36), where C_{k} = C_{k}(μx̃^{1} − 2λx̃^{3}) (k = 2, 3, 7) are arbitrary functions of one variable and the others are expressed from formulae

\begin{matrix}C_{1} = 2λ^{2} (μC′_{3} − C′_{7}) , C_{4} = −C_{1} − C_{3}, C_{5} = A + μC_{1}, \\ C_{6} = B, C_{8} = C_{5} + C_{7} (A, B = const) \end{matrix}  (144)

(the transformation of the coordinates is defined by formulae (41), the prime denotes differentiation).

Let C_{2} = Φ(μx̃^{1} − 2λx̃^{3}), and C_{3} = C_{7} = A = B = 0; taking into account the change (41), we obtain

F_{12} = F_{14} = F_{24} = 0, F_{13} = Φ(u), F_{23} = −F_{34} = \frac{x^{2} + x^{4}}{λ} Φ(u),  (145)

where u = μx̃^{1} − 2λx̃^{3} = 2λ(μx^{1} − x^{3}) + (x^{2} + x^{4})^{2}.

Proposition 68. If Φ′(u) ≠ 0, then the Maxwell space defined by the tensor F_{ij} of the form (144) admits the three-dimensional group G_{S} = G_{3,10}.

7.2.4. The class C_{3,10a}. The algebra L_{3,10a} = L{e_{12} − e_{14}, e_{1} + μe_{3}, e_{2} − e_{4}} (L_{3,10} for λ = 0 and μ ≠ 0) is an extension of the algebra L_{2,7a} = L{e_{12} − e_{14}, e_{2} − e_{4}} that corresponds to the class C_{2,7a}; thus C_{3,10a} ⊂ C_{2,7a}. The Maxwell space of the class C_{3,10a} is defined by the tensor F_{ij} of the form (36), where

\begin{matrix}C_{1} = \frac{a_{1}}{μx̃^{1}} + a_{2}, C_{2} = \frac{x̃^{3}C_{1} − b_{1}(x̃^{1})}{μx̃^{1}}, C_{3} = \frac{x̃^{3}C_{2}}{μx̃^{1}} + b_{2}(x̃^{1}), \\ C_{4} = −C_{1} − C_{3}, C_{5} = \frac{a_{1}}{x̃^{1}} , C_{6} = \frac{a_{1}x̃^{3}}{μ (x̃^{1})^{2}} b′_{1}(x̃^{1}), \\ C_{7} = −\frac{a_{1} (x̃^{3})^{2}}{2μ^{2} (x̃^{1})^{3}} − \frac{x̃^{3}b′_{1}(x̃^{1})}{μx̃^{1}} + b_{3}(x̃^{1}), C_{8} = C_{5} + C_{7}, \end{matrix}  (146)

a_{1}, a_{2} = const, b_{1}(x̃^{1}), b_{2}(x̃^{1}) and b_{3}(x̃^{1}) are arbitrary functions (the transformation of coordinates is defined by formulae (35), prime denotes differentiation with respect to x̃^{1}).

Let in (146) a_{1} = 0, a_{2} = A, b_{1} = b_{2} = b_{3} = 0; taking into account the change (35), we obtain

\begin{matrix}F_{12} = F_{14} = F_{24} = 0, F_{13} = \frac{A}{μ} \frac{x^{3} − μx^{1}}{x^{2} + x^{4}} , \\ F_{23} = \frac{A}{2μ^{2}} \frac{(x^{3} − μx^{1})^{2}}{(x^{2} + x^{4})^{2}} , F_{34} = −F_{23} − A. \end{matrix}  (147)

Proposition 69. If A ≠ 0, then the Maxwell space defined by the tensor F_{ij} of the form (147) admits the three-dimensional group G_{S} = G_{3,10a}.

7.2.5. The class C_{3,15}. The algebra L_{3,15} = L{e_{12} − e_{14}, e_{24}, e_{3}} is an extension of the algebra L_{2,12a}, hence C_{3,15} ⊂ C_{2,12a}. The Maxwell space of the class C_{3,15} is defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} = −\frac{C_{5}}{2} (x̃^{2})^{2} − C_{6}x̃^{2} + C_{7}, F_{13} = 0, F_{24} = C_{5}x̃^{2} + C_{6}, \\ F_{14} = −\frac{C_{5}}{2} (x̃^{2})^{2} − C_{6}x̃^{2} + C_{8}, F_{23} = −F_{34} = \frac{A}{x̃^{1}} (A = const), \end{matrix}  (148)

where the functions C_{k} = C_{k}(x̃^{1}, x̃^{4}) (k = 5, …, 8) satisfy equation (38d), the second equation in (37), and the system

\begin{matrix}x̃^{1} \frac{∂C_{5}}{∂x̃^{1}} + (x̃^{1})^{2} \frac{∂C_{5}}{∂x̃^{4}} − C_{5} = 0, \\ x̃^{1} \frac{∂C_{6}}{∂x̃^{1}} + (x̃^{1})^{2} \frac{∂C_{6}}{∂x̃^{4}} = 0, \\ x̃^{1} \frac{∂C_{7}}{∂x̃^{1}} + (x̃^{1})^{2} \frac{∂C_{7}}{∂x̃^{4}} + C_{8} = 0; \end{matrix}  (149)

the transformation of coordinates is defined by formulae (35).

Let the function Φ in (136) be independent of x^{3}; then we get instead (136) the tensor F_{ij} of the form

\begin{matrix}F_{12} = −\frac{1}{x^{2} + x^{4}} Φ − (\frac{2(x^{1})^{2}}{x^{2} + x^{4}} + 2(x^{2})^{2}) Φ′, F_{13} = F_{23} = F_{34} = 0, \\ F_{14} = −\frac{1}{x^{2} + x^{4}} Φ − (\frac{2(x^{1})^{2}}{x^{2} + x^{4}} − 2(x^{4})^{2}) Φ′, F_{24} = −2x^{1}Φ′, \end{matrix}  (150)

where Φ = Φ(t) = Φ((x^{1})^{2} + (x^{2})^{2} − (x^{4})^{2}) is an arbitrary function.

Proposition 70. If the functions Φ(t) and Φ′(t) are linearly independent, then the Maxwell space defined by the tensor (150) admits the three-dimensional group G_{S} = G_{3,15}.

7.2.6. The class C_{3,17}. The algebra L_{3,17} = L{e_{12} −e_{14}, e_{23} +e_{34}, e_{24}} is an extension of the algebra L_{2,12a}, hence C_{3,17} ⊂ C_{2,12a}. The Maxwell space of the class C_{3,17} is defined by the tensor F_{ij} of the form (36), where

\begin{matrix}C_{1} = \frac{Ay_{1}}{y_{4}^{2}} , C_{3} = −\frac{A (y_{1}^{2} + y_{3}^{2})}{2y_{1}y_{4}^{2}} − B \frac{y_{3}}{y_{1}} , C_{4} = −C_{1} − C_{3}, \\ C_{6} = \frac{Ay_{3}}{y_{4}^{2}} + B, C_{2} = C_{5} = C_{7} = C_{8} = 0 (A, B = const), \end{matrix}  (151)

and

y_{1} = x̃^{1}, y_{3} = x̃^{3}, y_{4} = x̃^{4} − \frac{1}{2} (x̃^{1})^{2} + \frac{1}{2} (x̃^{3})^{2} ;

the transformation of coordinates is defined by formulae (35).

The potential of the form

A_{i} = (Φ, −\frac{x^{1}}{x^{2} + x^{4}} Φ, 0, −\frac{x^{1}}{x^{2} + x^{4}} Φ) ,
Φ = Φ(t) = Φ((x^{1})^{2} + (x^{2})^{2} + (x^{3})^{2} − (x^{4})^{2})

belongs to the class P_{3,17}; it generates the following tensor F_{ij}:

\begin{matrix}F_{12} = −\frac{1}{x^{2} + x^{4}} Φ − (\frac{2(x^{1})^{2}}{x^{2} + x^{4}} + 2x^{2}) Φ′, \\ F_{14} = −\frac{1}{x^{2} + x^{4}} Φ − (\frac{2(x^{1})^{2}}{x^{2} + x^{4}} − 2x^{4}) Φ′, \\ F_{13} = −2x^{3}Φ′, F_{24} = −2x^{1}Φ′, F_{23} = −F_{34} = \frac{2x^{1}x^{3}}{x^{2} + x^{4}} Φ′. \end{matrix}  (152)

Proposition 71. If Φ′′(t) ≠ 0, then the Maxwell space defined by the tensor (152) admits the three-dimensional group G_{S} = G_{3,17}.

7.2.7. The class C_{3,18}. The algebra L_{3,18} = L{e_{12} − e_{14}, e_{23} + e_{34}, e_{13} + λ(e_{2} − e_{4})} is an extension of the algebra L_{2,11a}, thus C_{3,18} ⊂ C_{2,11a}. The Maxwell space of the class C_{3,18} is defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} = F_{14} = \frac{x^{1}}{x^{2} + x^{4}} Φ(x^{2} + x^{4}), F_{24} = Φ(x^{2} + x^{4}), \\ F_{23} = −F_{34} = −\frac{x^{3}}{x^{2} + x^{4}} Φ(x^{2} + x^{4}), F_{13} = 0, \end{matrix}  (153)

where Φ(t) is an arbitrary function.

Proposition 72. If Φ′(t) ≠ 0, then the Maxwell space defined by the tensor (153) admits the three-dimensional group G_{S} = G_{3,18}.

7.3. Four- and five-dimensional subgroups.

7.3.1. The class C_{4,9}. The algebra L_{4,9} = L{e_{12}−e_{14}+λe_{2}, e_{1}, e_{3}, e_{2}−e_{4}} is an extension of the algebra L_{3,8}, hence C_{4,9} ⊂ C_{3,8}. The Maxwell space of the class C_{4,9} is defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} = −\frac{b_{3}}{2} (x̃^{2})^{2} − b_{4}x̃^{2} + b_{5}, F_{13} = b_{1}, F_{14} = F_{12} + b_{3}, \\ F_{23} = −F_{34} = b_{1}x̃^{2} + b_{2}, F_{24} = b_{3}x̃^{2} + b_{4} (b_{k} = const), \end{matrix}  (154)

where x̃^{2} = (x^{2} + x^{4})/λ.

Let us take in (154) b_{3} = B = const, b_{1} = b_{2} = b_{4} = b_{5} = 0:

\begin{matrix}F_{12} = −\frac{B}{2λ^{2}} (x^{2} + x^{4})^{2} , F_{13} = F_{23} = F_{34} = 0, \\ F_{14} = F_{12} + B, F_{24} = \frac{B}{λ} (x^{2} + x^{4}). \end{matrix}  (155)

Proposition 73. If B ≠ 0, then the Maxwell space defined by the tensor (155) admits the four-dimensional group G_{S} = G_{4,9}.

7.3.2. The class C_{4,9a}. The algebra L_{4,9a} = L{e_{12} − e_{14}, e_{1}, e_{3}, e_{2} − e_{4}} is an extension of the algebra L_{2,7} (for λ = μ = 0), thus C_{4,9a} ⊂ C_{2,7a}. The Maxwell space of the class C_{4,9a} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = Φ(x^{2} + x^{4}), F_{13} = F_{24} = 0, F_{23} = −F_{34} = Ψ(x^{2} + x^{4}),  (156)

where Φ(u) and Ψ(u) are arbitrary functions.

Proposition 74. If the functions Φ, Ψ and (x^{2} + x^{4})Φ′ are linearly independent, then the Maxwell space defined by the tensor (156) admits the five-dimensional group G_{S} that corresponds to the algebra

L{e_{12} − e_{14}, e_{23} + e_{34}, e_{1}, e_{3}, e_{2} − e_{4}}.  (157)

There are no Maxwell’s spaces with the symmetry group G_{S} = G_{4,9a}.

7.3.3. The classes C_{4,12a} and C_{4,12b}. The algebra

L_{4,12} = L{e_{12} − e_{14} + μe_{3}, e_{23} + e_{34} + νe_{2}, e_{1}, e_{2} − e_{4}}

is an extension of the algebra L_{3,9} (for λ = 0). Therefore, the corresponding classes (for various μ and ν) are contained in C_{3,9a} or C_{3,9b}.

a) μ ≠ 0, ν = 0. The Maxwell space of the class C_{4,12a} that corresponds to the algebra L_{4,12a} = L{e_{12} − e_{14} + μe_{3}, e_{23} + e_{34}, e_{1}, e_{2} − e_{4}} (L_{4,12} for ν = 0) is defined by the tensor F_{ij} of the form (156).

Proposition 75. There are no Maxwell’s spaces with the symmetry group G_{S} = G_{4,12a}.

b) μ = 0, ν ≠ 0. The Maxwell space of the class C_{4,12b} that corresponds to the algebra L_{4,12b} = L{e_{12} − e_{14}, e_{23} + e_{34} + νe_{2}, e_{1}, e_{2} − e_{4}} (L_{4,12} for μ = 0) is defined by the tensor F_{ij} of the form.

\begin{matrix}F_{12} = F_{14} = A = const, F_{13} = F_{24} = 0, \\ F_{34} = −F_{23} = Ψ (x^{3} − \frac{1}{2ν} (x^{2} + x^{4})^{2}) . \end{matrix}  (158)

where Ψ(t) is an arbitrary function.

Proposition 76. If A ≠ 0 and Ψ′(t) ≠ 0, then the Maxwell space defined by the tensor (158) admits the four-dimensional group G_{S} = G_{4,12b}.

c) μ ≠ 0 and ν ≠ 0. In this case, the algebra is L_{4,12} and the class of the Maxwell spaces C_{6,3} is defined by (72).

Proposition 77. There are no Maxwell’s spaces with the symmetry group G_{S} = G_{4,12c} (G_{4,12} for μ ≠ 0 and ν ≠ 0).

7.3.4. The class C_{4,20}. The algebra L_{4,20} = L{e_{12}−e_{14}, e_{23}+e_{34}, e_{13}, e_{24}} is an extension of the algebra L_{3,17}, thus C_{4,20} ⊂ C_{3,17}. The Maxwell space of the class C_{4,20} is defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} = F_{14} = \frac{Bx^{1}}{x^{2} + x^{4}} , F_{23} = −F_{34} = −\frac{Bx^{3}}{x^{2} + x^{4}} , \\ F_{13} = 0, F_{24} = B (B = const ≠ 0). \end{matrix}  (159)

Proposition 78. The Maxwell space defined by the tensor F_{ij} of the form (159) admits the five-dimensional group G_{S} that corresponds to the algebra

L_{5,9} =

L{e_{12} − e_{14}, e_{23} + e_{34}, e_{13}, e_{24}, e_{2} − e_{4}}.

There are no Maxwell’s spaces with the symmetry group G_{S} = G_{4,20}.

7.3.5. The class C_{5,5}. The algebra L_{5,5} = L{e_{12} − e_{14}, e_{23} + e_{34} + λe_{2}, e_{2} − e_{4}, e_{1}, e_{3}} is an extension of the algebra L_{4,9a}, thence C_{5,5} ⊂ C_{4,9a}.

Proposition 79. For λ = 0 the class of the Maxwell spaces C_{5,5} coincides with C_{4,9a} and is defined by (156); for λ ≠ 0 it coincides with C_{6,3} and is defined by (72).

8. The classes of the Maxwell spaces that admit proportional bi-rotations

In this section, we describe classes of the Maxwell spaces that admit proportional bi-rotations and were not included in previous sections. These classes were found by E. G. Morokhova [19]. We use the change defined in Eq. (43).

8.1. Two-dimensional subgroups.

8.1.1. The class C_{2,9}. The algebra L_{2,9} = L{e_{13} + λe_{24}, e_{2} − e_{4}} corresponds to the group G_{2,9} generated by proportional bi-rotations and by translations along an isotropic straight line. As L_{1,5} ⊂ L_{2,9}, then the class C_{2,9} is a subclass of the class C_{1,5}. The Maxwell space of the class C_{2,9} is defined by the tensor F_{ij} of the form (44) – (45) if the following equations

\begin{matrix}\frac{∂F_{13}}{∂ρ} − \frac{1}{λρ} \frac{∂F_{13}}{∂θ} = 0, \frac{∂}{∂ρ}(c_{1} + c_{2}) = \frac{∂}{∂ρ}(c_{3} + c_{4}) = 0, \\ \frac{∂c_{1}}{∂ρ} − \frac{1}{λρ} \frac{∂c_{1}}{∂θ} − \frac{c_{2}}{ρ} − \frac{c_{3}}{λρ} = 0, \frac{∂c_{2}}{∂ρ} − \frac{1}{λρ} \frac{∂c_{2}}{∂θ} − \frac{c_{1}}{ρ} − \frac{c_{4}}{λρ} = 0, \\ \frac{∂c_{3}}{∂ρ} + \frac{1}{λρ} \frac{∂c_{3}}{∂θ} + \frac{c_{4}}{ρ} − \frac{c_{1}}{λρ} = 0, \frac{∂c_{4}}{∂ρ} + \frac{1}{λρ} \frac{∂c_{4}}{∂θ} + \frac{c_{3}}{ρ} − \frac{c_{2}}{λρ} = 0. \end{matrix}  (160)

are satisfied.

We may define the part of the class P_{2,9} of potentials A_{i} as follows,

A_{1} = A_{3} = 0, A_{2} = −A_{4} = ρΦ(r, λθ + ln ρ),  (161)

where Φ is an arbitrary function of two variables. For the case Φ = Φ(r) we obtain the tensor F_{ij} of the form

F_{12} = −F_{14} = ρe^{λφ}Φ′(r) cos(θ − φ), F_{13} = 0,

F_{23} = F_{34} = −ρe^{λφ}Φ′(r) sin(θ − φ), F_{24} = 2Φ(r). (162)

Proposition 80. If Φ′(r) ≠ 0, then the Maxwell space defined by the tensor (162) admits the two-dimensional group G_{S} = G_{2,9}.

8.1.2. The class C_{2,10}. The algebra L_{2,10} = L{e_{13}, e_{24}} = L{e_{13}+λe_{24}, e_{24}} corresponds to the group G_{2,10} generated by proportional bi-rotations and pseudo-rotations. As L_{1,5} ⊂ L_{2,10}, then C_{2,10} ⊂ C_{1,5}. The Maxwell space of the class C_{2,10} is defined by the tensor F_{ij} of the form (44) and

F_{13} = F_{13}(ρ, r), F_{24} = F_{24}(ρ, r),  (163)

where

c_{1} = −k_{3} sin θ + k_{4} cos θ, c_{2} = −k_{1} sin θ + k_{2} cos θ,
c_{3} = k_{3} cos θ + k_{4} sin θ, c_{4} = k_{1} cos θ + k_{2} sin θ,

(164) and the functions F_{13}(ρ, r), F_{24}(ρ, r), and k_{i} = k_{i}(ρ, r) satisfy the equations

\frac{∂F_{24}}{∂r} + \frac{∂k_{1}}{∂ρ} + \frac{k_{1}}{ρ} = 0, \frac{∂F_{13}}{∂ρ} − \frac{∂k_{4}}{∂r} − \frac{k_{4}}{r} = 0, k_{2} = \frac{K}{rρ} (K = const).  (165)

The part of the class P_{2,10} of potentials that admits the group G_{2,10} is defined as follows:

A_{i} = (0, Φ(r, ρ)e^{λφ}, 0, −Φ(r, ρ)e^{λφ}),

where Φ = Φ(r, ρ) is an arbitrary function. This potential generates the tensor F_{ij} such that

F_{12} = −F_{14} = e^{λφ}Φ′_{r}(r, ρ) cos(θ − φ), F_{13} = 0,

F_{23} = F_{34} = −e^{λφ}Φ′_{r}(r, ρ) sin(θ − φ), F_{24} = − (Φ′_{ρ}(r, ρ) + \frac{1}{ρ}Φ(r, ρ)) . (166)

Proposition 81. If the conditions

Φ′′_{rρ} + \frac{1}{ρ}Φ′_{r} ≠ 0, Φ′′_{ρρ} + \frac{1}{ρ}Φ′_{ρ} − \frac{1}{ρ^{2}} Φ ≠ 0,

are satisfied, then the Maxwell space defined by the tensor in (166) admits the two-di-mensional group G_{S} = G_{2,10}.

8.2. Three-dimensional subgroups.

8.2.1. The class C_{3,11}. The algebra L_{3,11} = L{e_{13} + λe_{24}, e_{1}, e_{3}} corresponds to the group G_{3,11} generated by proportional bi-rotations and by translations along the vectors of two-dimensional Euclidian plane. The Maxwell space of the class C_{3,11} is defined by the tensor F_{ij} of the form (44) and

F_{13} = const, F_{24} = F_{24}(ρ),  (167)

where F_{24}(ρ), c_{2} = c_{2}(ρ), and c_{4} = c_{4}(ρ) are arbitrary functions and

c_{1} = λ(c_{4} + ρc′_{4}), c_{3} = −λ(c_{2} + ρc′_{2}).  (168)

Let us take in (167), (168), and (44) c_{2} = Φ(ρ), c_{4} = 0, F_{24} = Ψ(ρ); we get an example of the tensor F_{ij} that defines the Maxwell space of the class C_{3,11}:

F_{12} = −Φ(ρ) sinh λφ sin φ − λ(Φ(ρ) + ρΦ′(ρ)) cosh λφ cos φ,
F_{14} = Φ(ρ) cosh λφ sin φ + λ(Φ(ρ) + ρΦ′(ρ)) sinh λφ cos φ,
F_{23} = Φ(ρ) sinh λφ cos φ − λ(Φ(ρ) + ρΦ′(ρ)) cosh λφ sin φ,
F_{34} = Φ(ρ) cosh λφ cos φ − λ(Φ(ρ) + ρΦ′(ρ)) sinh λφ sin φ,
F_{13} = const, F_{24} = Ψ(ρ).

(169)

Proposition 82. If Φ(ρ) ≠ const and Ψ(ρ) ≠ const, then the Maxwell space defined by the tensor (169) admits the three-dimensional group G_{S} = G_{3,11}.

8.2.2. The class C_{3,12}. The algebra L_{3,12} = L{e_{13} + λe_{24}, e_{2}, e_{4}} corresponds to the group G_{3,12} generated by proportional bi-rotations and by translations along the vectors of two-dimensional pseudo-Euclidian plane. The Maxwell space of the class C_{3,12} is defined by the tensor F_{ij} of the form (44) and

F_{13} = F_{13}(r), F_{24} = const,  (170)

where

c_{1} = sin θ(t_{1} cosh λθ + t_{2} sinh λθ) − cos θ(t_{3} cosh λθ + t_{4} sinh λθ),
c_{2} = − sin θ(t_{1} sinh λθ + t_{2} cosh λθ) + cos θ(t_{3} sinh λθ + t_{4} cosh λθ),
c_{3} = − cos θ(t_{1} cosh λθ + t_{2} sinh λθ) − sin θ(t_{3} cosh λθ + t_{4} sinh λθ),
c_{4} = cos θ(t_{1} sinh λθ + t_{2} cosh λθ) + sin θ(t_{3} sinh λθ + t_{4} cosh λθ),

(171)

moreover, the functions F_{13}(r), t_{3} = t_{3}(r), and t_{4} = t_{4}(r) are arbitrary and

t_{1} = −\frac{1}{λ}(t_{4} + rt′_{4}), t_{2} = −\frac{1}{λ}(t_{3} + rt′_{3}).  (172)

Proposition 83. If F_{13} ≠ const and t_{3} ≠ const (or t_{4} ≠ const), then the Maxwell space defined by the tensor F_{ij} according to formulae (44), (170), (171), and (172) admits the three-dimensional group G_{S} = G_{3,12}.

8.2.3. The class C_{3,13}. The algebra L_{3,13} = L{e_{13}, e_{24}, e_{2} − e_{4}} is an extension of the algebra L_{2,10}, thus C_{3,13} ⊂ C_{2,10}. The Maxwell space of the class C_{3,13} is defined by the tensor F_{ij} of the form (44), (164) and

F_{13} = Φ_{1}(r), F_{24} = Φ_{2}(r),  (173)

where

k_{1} = −\frac{1}{2}Φ′_{2}(r)ρ^{2} + Φ_{3}(r), k_{2} = −k_{4} = \frac{K}{rρ}, k_{3} = −\frac{1}{2}Φ′_{2}(r)ρ^{2} − Φ_{3}(r)  (174)

(K = const, Φ_{1}(r), Φ_{2}(r), Φ_{3}(r) are arbitrary functions).

8.3. Four-dimensional subgroups.

8.3.1. The class C_{4,7}. The algebra L_{4,7} = L{e_{13} + λe_{24}, e_{1}, e_{3}, e_{2} + e_{4}} is an extension of the algebra L_{3,11}, hence C_{4,7} ⊂ C_{3,11}. The Maxwell space of the class C_{4,7} is defined by the tensor F_{ij} of the form

−F_{12} = F_{14} = (c_{1} sin φ − c_{2} cos φ) e^{λφ}, F_{13} = B = const,
F_{23} = F_{34} = (c_{1} cos φ + c_{2} sin φ) e^{λφ}, F_{24} = E = const,

(175)

where c_{1} = c_{1}(ρ), c_{2} = c_{2}(ρ) are arbitrary functions).

Proposition 84. Suppose two following conditions are satisfied:

1) c_{1} ≠ const or c_{2} ≠ const,

2) B ≠ 0 or E ≠ 0. Then the Maxwell space defined by the tensor F_{ij} of the form (175) admits the four-di-mensional group G_{S} = G_{4,7}.

8.3.2. The class C_{4,10}. The algebra L_{4,10} = L{e_{13}, e_{24}, e_{1}, e_{3}, } contains L_{2,10} as a subalgebra, therefore the class C_{4,10} is a subclass of C_{2,10}. The Maxwell space of the class C_{4,10} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{23} = F_{34} = 0, F_{13} = B = const, F_{24} = F_{24}(ρ)  (176)

(F_{24}(ρ) is an arbitrary function).

Proposition 85. If B ≠ 0 and F_{24}(ρ) ≠ const, then the Maxwell space defined by the tensor F_{ij} of the form (176) admits the four-dimensional group G_{S} = G_{4,10}.

8.3.3. The class C_{4,11}. The algebra L_{4,11} = L{e_{13}, e_{24}, e_{2}, e_{4}} is also an extension of the algebra L_{2,10}, hence C_{4,11} ⊂ C_{2,10}. The Maxwell space of the class C_{4,11} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} = F_{23} = F_{34} = 0, F_{13} = F_{13}(r), F_{24} = E = const.  (177)

(F_{13}(r) is an arbitrary function).

Proposition 86. If E ≠ 0 and F_{13}(r) ≠ const, then the Maxwell space, defined by the tensor F_{ij} of the form (177) admits the four-dimensional group G_{S} = G_{4,11}.

9. Other classes

In this section, we describe classes of Maxwell spaces which were not described in the previous sections.

9.0.4. The classes C_{3,14a}, C_{3,14b} and C_{3,14c}. The algebra

L_{3,14} = L{e_{12} − e_{14} + λe_{1} + μe_{3}, e_{23} + e_{34} + νe_{1} + λe_{3}, e_{2} − e_{4}}

corresponds to the group G_{3,14} generated by two one-dimensional subgroups of parabolic helices and by translations along an isotropic straight line. We describe the corresponding classes in three particular cases.

a) λ = μ = ν = 0. The Maxwell space of the class C_{3,14a} that corresponds to the algebra L_{3,14a} = L{e_{12} − e_{14}, e_{23} + e_{34}, e_{2} − e_{4}} is defined by the tensor F_{ij} of the form

\begin{aligned}F_{13} & = \frac{K_{1}x^{1} + K_{2}x^{3} + K_{3}}{(x^{2} + x^{4})^{2}} , F_{24} = \frac{K_{2}x^{1} − K_{1}x^{3}}{(x^{2} + x^{4})^{2}} + Φ_{1}(x^{2} + x^{4}), \\ F_{14} & = \frac{K_{2} ((x^{1})^{2} − (x^{3})^{2}) − 2K_{1}x^{1}x^{3} − 2K_{3}x^{3}}{2 (x^{2} + x^{4})^{3}} + \frac{x^{1}Φ_{1}(x^{2} + x^{4})}{x^{2} + x^{4}}+ \\ & + Φ_{2}(x^{2} + x^{4}), F_{12} = F_{14} + \frac{K_{2}}{x^{2} + x^{4}} , \\ F_{34} & = \frac{K_{1} ((x^{1})^{2} − (x^{3})^{2}) + 2K_{2}x^{1}x^{3} + 2K_{3}x^{1}}{2 (x^{2} + x^{4})^{3}} + \frac{x^{3}Φ_{1}(x^{2} + x^{4})}{x^{2} + x^{4}}+ \\ & + Φ_{3}(x^{2} + x^{4}), F_{23} = −F_{34} + \frac{K_{1}}{x^{2} + x^{4}} , \end{aligned}  (178)

where K_{1}, K_{2}, K_{3} = const, and Φ_{1}(t), Φ_{2}(t), and Φ_{3}(t) are arbitrary functions of one variable.

b) λ ≠ 0 and μ = ν = 0. The Maxwell space of the class C_{3,14b} that corresponds to the algebra L_{3,14b} = L{e_{12} − e_{14} + λe_{1}, e_{23} + e_{34} + λe_{3}, e_{2} − e_{4}} is defined by the tensor F_{ij} of the form

\begin{matrix}F_{12} & = F_{14} + \frac{K_{2}}{x^{2} + x^{4} − λ}, F_{13} = \frac{K_{1}x^{1} + K_{2}x^{3} + K_{3}}{(x^{2} + x^{4})^{2} − λ^{2}} , \\ F_{14} & = \frac{K_{2} (x^{1})^{2}}{2 (x^{2} + x^{4} − λ)^{3}} − \frac{2K_{1}x^{1}x^{3} + K_{2} (x^{3})^{2} + 2K_{3}x^{3}}{2 (x^{2} + x^{4} + λ)^{2} (x^{2} + x^{4} − λ)}+ \\ + \frac{x^{1}Φ_{1}(x^{2} + x^{4})}{x^{2} + x^{4} − λ} + Φ_{2}(x^{2} + x^{4}), \\ F_{24} & = \frac{K_{2}x^{1}}{(x^{2} + x^{4} − λ)^{2}} − \frac{K_{1}x^{3}}{(x^{2} + x^{4} + λ)^{2}} + Φ_{1}(x^{2} + x^{4}), \\ F_{34} = −\frac{K_{1} (x^{3})^{2}}{2 (x^{2} + x^{4} + λ)^{3}} + \frac{K_{1} (x^{1})^{2} + 2K_{2}x^{1}x^{3} + 2K_{3}x^{1}}{2 (x^{2} + x^{4} − λ)^{2} (x^{2} + x^{4} + λ)}+ \\ + & \frac{x^{3}Φ_{1}(x^{2} + x^{4})}{x^{2} + x^{4} + λ} + Φ_{3}(x^{2} + x^{4}), F_{23} = −F_{34} + \frac{K_{1}}{x^{2} + x^{4} + λ}. \end{matrix}  (179)

Formulae (179) transforms to (178) if λ = 0.

c) λ = 0 and μ = ν ≠ 0. The Maxwell space of the class C_{3,14c} that corresponds to the algebra L_{3,14c} = L{e_{12} − e_{14} + μe_{3}, e_{23} + e_{34} + μe_{1}, e_{2} − e_{4}} is defined by the tensor F_{ij} of the form

F_{12} = F_{14} + μΨ_{1}(u) + uΨ_{2}(u), F_{13} = x^{1}Ψ_{1}(u) + x^{3}Ψ_{2}(u) + C_{1}(u),
F_{23} = −F_{34} + uΨ_{1}(u) − μΨ_{2}(u), F_{24} = x^{1}Ψ_{2}(u) − x^{3}Ψ_{1}(u) + C_{2}(u),
F_{14} = F_{14}(x^{1}, x^{3}, u), F_{34} = F_{34}(x^{1}, x^{3}, u) (u = x^{2} + x^{4}),

(180)

where

\begin{matrix}Ψ_{1}(u) = \frac{(K_{1} + 2K_{2}μ) (u^{2} − μ^{2}) − 2K_{3}μu}{(u^{2} + μ^{2})^{2}}, \\ Ψ_{2}(u) = −\frac{2K_{1}μu + 4K_{2}μ^{2}u + K_{3} (u^{2} − μ^{2})}{(u^{2} + μ^{2})^{2}}, \\ C_{1}(u) = \frac{K_{4} − 2μΦ(u)}{u^{2} + μ^{2}}, C_{2}(u) = Φ′(u) \end{matrix}  (181)

(K_{1}, …, K_{4} = const, Φ(u) is an arbitrary function), and the components F_{14}, F_{34} satisfy the equations

\begin{matrix}∂_{1}F_{14} = ∂_{3}F_{34} = −\frac{μF_{13} − uF_{24}}{u^{2} + μ^{2}} , \\ ∂_{3}F_{14} = −∂_{1}F_{34} = \frac{uF_{13} + μF_{24}}{u^{2} + μ^{2}} . \end{matrix}  (182)

9.0.5. The class C_{3,20}. The algebra L_{3,20} = L{e_{12}, e_{13}, e_{23}} corresponds to the three-di-mensional group G_{3,20} of rotations around the origin in the subspace of the Minkowski space ℝ_{0}^{3} = {x ∈ ℝ_{1}^{4} : x^{4} = 0}. The Maxwell space of the class C_{3,20} is defined by the tensor F_{ij} of the form

F_{12} = −Ax^{3}/r^{3}, F_{13} = Ax^{2}/r^{3}, F_{23} = −Ax^{1}/r^{3} (A = const),

F_{14} = −x^{1}Φ(r, x^{4}), F_{24} = −x^{2}Φ(r, x^{4}), F_{34} = −x^{1}Φ(r, x^{4}). (183)

9.0.6. The class C_{4,16}. For the algebra

L_{4,16} = L{e_{12} − e_{14} + λe_{3}, e_{23} + e_{34} + λe_{1}, e_{13}, e_{2} − e_{4}}

we obtain the empty class C_{4,16}.

9.0.7. The class C_{6,1}. The algebra L_{6,1} = L{e_{12}, e_{13}, e_{23}, e_{14}, e_{24}, e_{34}} corresponds to the Lorentz group G_{6,1}. The class of Maxwell spaces that corresponds to the Lorentz group is empty.

Acknowledgements. The author thanks A. V. Bolsinov and A. T. Fomenko for their constructive remarks, which helped to improve the formulation of the initial problem.

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Ivanovo State University. 153025 Russia, Ivanovo, Ermaka str., 39. IvSU, Chair of Geometry.

E-mail address: parinov@ivanovo.ac.ru