On Symmetries of Integrable Quadrilateral Equations

In the paper, we describe a method for deriving generalized symmetries for a generic discrete quadrilateral equation that allows a Lax pair. Its symmetry can be interpreted as a flow along the tangent direction of its solution evolving with a Lie group parameter t. Starting from the spectral problem of the quadrilateral equation and assuming the eigenfunction evolves with the parameter t, one can obtain a differential-difference equation hierarchy, of which the flows are proved to be commuting symmetries of the quadrilateral equation. We prove this result by using the zero-curvature representations of these flows. As an example, we apply this method to derive symmetries for the lattice potential Korteweg–de Vries equation.