A 2x2 Lax representation, associated family, and B\"acklund transformation for circular K-nets

We present a 2x2 Lax representation for discrete circular nets of constant negative Gau{\ss} curvature. It is tightly linked to the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to B\"acklund transformations and an associated family. All the members of that family -- although no longer circular -- can be shown to have constant Gau{\ss} curvature as well. Explicit solutions for the B\"acklund transformations of the vacuum (in particular Dini's surfaces and breather solutions) and their respective associated families are given.