Generalised Bianchi permutability for isothermic surfaces

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

Automated summary

This paper discusses the Darboux transforms and their properties for isothermic surfaces, focusing on the two-step transforms with the same spectral parameter. It provides a method for calculating these transforms using parallel sections of the associated family of the isothermic surface, without the need for further integration.