Asymptotic lattices and their integrable reductions: I. The Bianchi-Ernst and the Fubini-Ragazzi lattices

We review recent results on asymptotic lattices and their integrable reductions. We present the theory of general asymptotic lattices in R-3 together with the corresponding theory of their Darboux-type transformations. Then we find a novel permutability theorem for Bianchi surfaces, which can be reinterpreted as a discrete version of the Bianchi-Ernst system and coincides with an equation recently introduced by Schief (Schief W K 2001 Stud. Appl. Math. 106 85-137). Using the well known connection between the Bianchi and Ernst systems, we also propose the discrete analogue of the Ernst system. Finally, we present the theory of the discrete analogues of isothermally asymptotic (Fubini-Ragazzi) nets together with their transformations.