Integration of the Gauss-Codazzi equations

The Gauss-Codazzi equations imposed on the elements of the first and the se cond quadratic forms of a surface embedded in R-3 are integrable by the dre ssing method. This method allows constructing classes of Combescure-equival ent surfaces with the same "rotation coefficients." Each equivalence class is defined by a function of two variables ("master function of a surface"). Each class of Combescure-equivalent surfaces includes the sphere. Differen t classes of surfaces define different systems of orthogonal coordinates of the sphere. The simplest class (with the master function zero) corresponds to the standard spherical coordinates.