Graded discrete Lax pairs and integrable difference equations

Abstract

We introduce a class of graded discrete Lax pairs, with matrices, linear in the spectral parameter. We give a classification scheme for such Lax pairs and the associated integrable lattice systems. We present two potential forms and completely classify the generic case. Many well known examples belong to our scheme for N = 2, so many of our systems may be regarded as generalisations of these. Even at N = 3, several new integrable systems arise. A decomposable case gives rise to interesting coupled systems of lower dimensional equations. Many of our equations are mutually compatible, so can be used together to form ‘coloured’ lattices.