Coverings and multivector pseudosymmetries of differential equations

Finite-dimensional coverings from systems of differential equations are investigated. This problem is of interest in view of its relationship with the computation of differential substitution, nonlocal symmetries, recursion operators, and Backlund transformations. We show that the distribution specified by the fibers of a covering is determined by an integrable pseudosymmetry of the system. Conversely, every integrable pseudosymmetry of a system defines a covering from this system. The vertical component of the pseudosymmetry is a matrix analog of the evolution differentiation. The corresponding generating matrix satisfies a matrix analog of the linearization of the equation. We consider also the exterior product of vector fields defining a pseudosymmetry. The definition of pseudosymmetry is rewritten in the language of the Schouten bracket of multivector fields and total derivatives with respect to the independent variables of the system. A method for constructing coverings is given and demonstrated by the examples of the Laplace equation and the Kapitsa pendulum system.