Group interpretation of the spectral parameter. The case of isothermic surfaces

Abstract

It is well known that in some cases the spectral parameter has a group interpretation. We discuss in detail the case of Gauss-Codazzi equations for isothermic surfaces immersed in $E^3$. The algebra of Lie point symmetries is 4-dimensional and all these symmetries are also symmetries of the Gaus-Weingarten equations (which can be considered as so(3)-valued non-parametric linear problem). In order to obtain a non-removable spectral parameter one has to consider so(4,1)-valued linear problem which has a 3-dimensional algebra of Lie point symmetries. The missing symmetry introduces a non-removable parameter. In the second part of the paper we extend these results on the case of isothermic immersions in arbitrary multidimensional Euclidean spaces. In order to simplify calculations the problem was formulated in terms of a Clifford algebra.