Generalised Bianchi permutability for isothermic surfaces

TL;DR

This paper explores the algebraic construction of two-step Darboux transforms for isothermic surfaces, extending Bianchi permutability to cases with the same spectral parameter using a Sym-type approach.

Contribution

It introduces a novel algebraic method to obtain two-step Darboux transforms with the same spectral parameter for isothermic surfaces, avoiding additional integration.

Findings

Two-step Darboux transforms are obtained via parallel sections of the associated family.

The construction can be performed algebraically or through differentiation with respect to the spectral parameter.

The approach extends Bianchi permutability to the case of identical spectral parameters.

Abstract

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.