On two integrable lattice equations and their interpretation

We examine two lattice equations, obtained through the application of multiscale perturbative analysis, from the point of view of integrability. We show that both equations are integrable and related to the discrete sine-Gordon. We compute the limit of both systems whereby they become linearizable, obtaining the discrete Liouville equation and a linearizable lattice system recently proposed by Hydon and Viallet. We present the explicit solution of the latter.