Generalised Bianchi permutability for isothermic surfaces

Abstract Isothermic surfaces are which allow a conformal curvature line parametrisation. They form an integrable system, and Darboux transforms of isothermic obey Bianchi permutability: for two distinct spectral parameters, the corresponding have common transform can be computed algebraically. In this paper, we discuss two-step with same parameter, show that these obtained by Sym-type construction: All surface given, without further integration, parallel sections associated family surface, either algebraically or differentiation against parameter.