Analogs of coverings and the fundamental group for the category of partial differential equations

We introduce a new geometric invariant of PDEs: with any analytic system of
PDEs we associate naturally a certain system of Lie algebras. Using
infinite jet spaces, one can regard PDEs as geometric objects (manifolds
with distributions) and obtains a category of PDEs. We study a special kind
of morphisms in this category: the Krasilshchik-Vinogradov coverings, which
generalize the classical concept of coverings from topology. They provide a
geometric framework for Backlund transformations, which are a well-known
tool to construct exact solutions for nonlinear PDEs. Recall that
topological coverings of a manifold M can be described in terms of the
fundamental group of M. We show that a similar description exists for
Krasilshchik-Vinogradov coverings of PDEs. However, the "fundamental group
of a PDE" is not a group, but a certain system of Lie algebras, which we
call fundamental algebras. In particular, these algebras are responsible
for Backlund transformations and Lax pairs in the theory of integrable
systems. We have computed these algebras for several well-known nonlinear
PDEs. As a result, one obtains infinite-dimensional Lie algebras of
Kac-Moody type and Lie algebras of matrix-valued functions on algebraic
curves. Applications to construction and classification of Backlund
transformations will be also presented.