Coverings and fundamental algebras for partial differential equations

Following I. S. Krasilshchik and A. M. Vinogradov, we regard PDEs as infinite-dimensional manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and Backlund transformations. We show that, similarly to usual coverings in topology, at least for some PDEs differential coverings are determined by actions of a sort of fundamental group. This is not a group, but a certain system of Lie algebras, which generalize Wahlquist-Estabrook algebras. From this we deduce an algebraic necessary condition for two PDEs to be connected by a Backlund transformation. We compute these infinite-dimensional Lie algebras for several KdV type equations and prove non-existence of Backlund transformations. As a by-product, for some class of Lie algebras g we prove that any subalgebra of g of finite codimension contains an ideal of g of finite codimension.