**DIPS 3/2002**[tex source, PostScript, PDF file, dvi]-
**Title:**Conservation laws for multidimensional systems and related linear algebra problems.**Author:**Sergei IGONINWe consider multidimensional systems of PDEs of generalized evolution form with t-derivatives of arbitrary order on the left-hand side and with the right-hand side dependent on lower order t-derivatives and arbitrary space derivatives. For such systems we find an explicit necessary condition for existence of higher conservation laws in terms of the system's symbol. For systems that violate this condition we give an effective upper bound on the order of conservation laws. Using this result, we completely describe conservation laws for viscous transonic equations, for the Brusselator model, and the Belousov-Zhabotinskii system. To achieve this, we determine the conditions for a quadratic matrix A with entries from an arbitrary field to be similar (conjugate) to its transpose A

^{t}or to the matrix -A^{t}, which is of independent interest.

11 pages, LaTeX-2e (AmSLaTeX).

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