Extension of the module of invertible transformations. Classification of integrable systems (Q1112261)

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 form \[ u_ t=u_{xx}+f(u,v,u_ x,v_ x),\quad -v_ t=v_{xx}+g(u,v,u_ x,v_ x), \] 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.