Coverings and the fundamental group for partial differential equations

Following I. S. Krasilshchik and A. M. Vinogradov, we regard systems of PDEs as manifolds with involutive distributions and consider their special morphisms called differential coverings, which include constructions like Lax pairs and B\"acklund 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 discrete group, but a certain system of Lie groups. From this we deduce an algebraic necessary condition for two PDEs to be connected by a B\"acklund transformation. For the KdV equation and the nonsingular Krichever-Novikov equation these systems of Lie groups are determined by certain infinite-dimensional Lie algebras of Kac-Moody type. We prove that these two equations are not connected by any B\"acklund transformation. To achieve this, for a wide class of Lie algebras $\mathfrak{g}$ we prove that any subalgebra of $\mathfrak{g}$ of finite codimension contains an ideal of $\mathfrak{g}$ of finite codimension.