Linearizability and a fake Lax pair for a nonlinear nonautonomous quad-graph equation consistent around the cube

Abstract

We discuss the linearization of a nonautonomous nonlinear partial difference equation belonging to the Boll classification of quad-graph equations consistent around the cube. We show that its Lax pair is fake. We present its generalized symmetries, which turn out to be nonautonomous and dependent on an arbitrary function of the dependent variables defined at two lattice points. These generalized symmetries are differential–difference equations, which admit peculiar Bäcklund transformations in some cases.