Coverings and the Fundamental Group for Partial Differential Equations

Following I. S. Krasilshchik and A. M. Vinogradov [8], 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 Bäcklund transformations in soliton theory. 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. From this we deduce an algebraic necessary condition for two PDEs to be connected by a Bäcklund transformation. We compute these Lie algebras for several integrable evolution PDEs including the KdV equation and prove non-existence of Bäcklund transformations. To achieve this, 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.