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.Comment: 50 pages. v3: considerable corrections; v4,v5: minor change

We completely describe Wahlquist-Estabrook prolongation structures (coverings) dependent on u, u x , u xx , u xxx for the Krichever-Novikov equation u t = u xxx −3u 2 xx /(2u x )+p(u)/u x +au x in the case when the polynomial p(u) = 4u 3 − g 2 u − g 3 has distinct roots. We prove that there is a universal prolongation algebra isomorphic to the direct sum of a commutative 2-dimensional algebra and a certain subalgebra of the tensor product of sl 2 (C) with the algebra of regular functions on an affine elliptic curve. This is achieved by identifying this prolongation algebra with the one for the anisotropic Landau-Lifshitz equation. Using these results, we find for the Krichever-Novikov equation a new zero-curvature representation, which is polynomial in the spectral parameter in contrast to the known elliptic ones. (2000): 37K10, 37K30, 35Q53 Mathematics Subject Classification

We study the equation E fc of flat connections in a given fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an equation E are in 1-1 correspondence with morphisms ϕ : E → E fc , where E and E fc are treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-curvature formulations of PDEs. In particular, the Lax pairs of the self-dual Yang-Mills equations and their reductions are of this type. With each flat representation ϕ we associate a complex C ϕ of vectorvalued differential forms such that H 1 (C ϕ ) describes infinitesimal deformations of the flat structure, which are responsible, in particular, for parameters in Bäcklund transformations. In addition, each higher infinitesimal symmetry S of E defines a 1-cocycle c S of C ϕ . Symmetries with exact c S form a subalgebra reflecting some geometric properties of E and ϕ. We show that the complex corresponding to E fc itself is 0-acyclic and 1-acyclic (independently of the bundle topology), which means that higher symmetries of E fc are exhausted by generalized gauge ones, and compute the bracket on 0-cochains induced by commutation of symmetries. Contents

We consider multidimensional systems of PDEs of generalized evolution form with t-derivatives of arbitrary order on the left-hand side and with the right-hand side dependent on lower order t-derivatives and arbitrary space derivatives. For such systems we find an explicit necessary condition for existence of higher conservation laws in terms of the system's symbol. For systems that violate this condition we give an effective upper bound on the order of conservation laws. Using this result, we completely describe conservation laws for viscous transonic equations, for the Brusselator model, and the Belousov-Zhabotinskii system. To achieve this, we solve over an arbitrary field the matrix equations SA = A t S and SA = −A t S for a quadratic matrix A and its transpose A t , which may be of independent interest. (2000): Primary 37K05, 37K10; Secondary 15A24, 76H05, 35K57. Mathematics Subject Classification