A completeness study on a class of discrete, 'two by two' Lax pairs

We propose a method by which to examine all possible partial difference Lax pairs that consist of 'two by two' discrete linear problems, where the matrices contain one separable term in each entry. We thereby derive new, higher-order versions of the lattice sine-Gordon and lattice modified KdV equations, while showing that there can be no other partial difference equations associated with this type of Lax pair.