Symmetries of Integrable Difference Equations on the Quad‐Graph

Abstract

This paper describes symmetries of all integrable difference equations that belong to the famous Adler–Bobenko–Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five‐point symmetries of integrable equations on the quad‐graph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quad‐graph equations and those of the corresponding Toda type difference equations.