Extension of the module of invertible transformations. Classification of integrable systems

Abstract

We demonstrate that for the systems of equations, which are invariant under a point group or possess conservation laws of the zeroth or first order, a nontrivial extension of the module of invertible transformations is possible. That simplifies greatly a classification of the integrable systems of equations. Here we present an exhaustive list and a classification of the second order systems of the formu t =u xx +f(u, v, u x v x ), −v t=v xx +g(u, v, u x ,v x ), which possess the conservation laws of higher order. The reduction group approach allows us to define the Lax type representations for some new equations of our list.