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

Abstract: We discuss the linearization of a non-autonomous 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 non-autonomous and depending on an arbitrary function of the dependent variables defined in two lattice points. These generalized symmetries are differential difference equations which, in some case, admit peculiar B\"acklund transformations.