**DIPS 4/2004**[tex source, PostScript, PDF file, dvi]-
**Title:**Towards classification of conditionally integrable evolution systems in (1+1) dimensions.**Author:**A. SergyeyevWe generalize earlier results of Fokas and Liu and find all locally analytic (1+1)-dimensional evolution equations of order n that admit an N-shock type solution with N< n+2. To this end, we develop a refinement of the technique from our earlier work (A. Sergyeyev,

*J. Phys. A: Math. Gen*,**35**(2002), 7653-7660), where we completely characterized all (1+1)-dimensional evolution systems that are conditionally invariant under a given generalized (Lie-Baecklund) vector field under the assumption that the system of ODEs**Q**=0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics.

11 pages, AmS-LaTeX (LaTeX-2e).

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