Determining the symmetries of difference equations

We derive the determining equations for the Nth-order generalized symmetries of partial difference equations defined on d consecutive quadrilaterals on the lattice using the theory of integrability conditions. We provide their algebraic formulation and develop the necessary theoretical framework for their analysis along with a systematic method for solving functional equations of the form T( f)+Af+B=0. Our approach is algorithmic and can be easily implemented in symbolic computations. We demonstrate our approach by deriving the symmetries of various equations and discuss certain applications and extensions of the theory.