Symmetry approach in differential and symbolic representations

Abstract:
Existence of higher symmetries is the characteristic property of integrable equations. It is a difficult and very delicate problem to describe a variety of equations which possess higher symmetries. The central object of the Symmetry Approach is the concept of formal recursion operator. Symbolic representation, suitable for differential polynomials enables to solve a global problem of classification and give a complete description of equations with symmetries in all orders. Our recent attempt to merge the Symmetry Approach and the Symbolic Method looks promising and suitable for non-evolutionary and multi-dimensional equations.