The Darboux-Bianchi transformation for isothermic surfaces. Classical results versus the soliton approach

AbstractWe show that theory of soliton surfaces, modified in an appropriate way, can be applied also to isothermic immersions in E3. In this case the so called Sym's formula gives an explicit expression both for the isothermic immersion with prescribed fundamental forms and its Christoffel transform (dual surface). Then, applying methods of the theory of solitons, we construct the Darboux matrix for isothermic immersions reconstructing in this way the classical Darboux-Bianchi transformation for isothermic surfaces