Inverse problem of the theory of compatibility and functional-invariant solutions wave equation in two-dimensional space

We investigated the system of equations with variable coefficients, which describefunctional-invariant solutions of wave equation in spaceR3(t,x,y). It is well known that inthe case of identity matrix of coefficients we can describe allfunctional-invariant solutionsby Sobolev formula. In this paper we prove that if solutions of considered systems havemaximal arbitrariness (in the sense of the theory of compatibility overdetermined systemsof differential equations in partial derivatives) then coefficients of the wave equation areconnected by algebraic relation of the second order (hyperbolic or elliptic type) and inaddition by differential relation of the second order. Groupof transformations induced bychange of space variables acts on the all set of differential equations naturally. We obtaincomplete classification of the considered systems with respect to this group. More precisely,we prove that there are exactly three classes of equivalence. In this paper we use classicalmethods Requier theory of investigation of overdeterminedsystems of differential equationsin partial derivatives.