Coverings and integrable pseudosymmetries of differential equations

Abstract

We study the problem on the construction of coverings by a given system of differential equations and the description of systems covered by it. This problem is of interest in view of its relationship with the computation of nonlocal symmetries, recursion operators, B¨acklund transformations, and decompositions of systems. We show that the distribution specified by the fibers of the covering is determined by a pseudosymmetry of the system and is integrable in the infinite-dimensional sense. Conversely, every integrable pseudosymmetry of a system defines a covering by this system. The vertical component of the pseudosymmetry is a matrix analog of the evolution differentiation, and the corresponding generating matrix satisfies a matrix analog of the linearization of an equation.