Integration of the Gauss–Codazzi Equations

The Gauss–Codazzi equations imposed on the elements of the first and the second quadratic forms of a surface embedded in $\mathbb R^3$ are integrable by the dressing method. This method allows constructing classes of Combescure-equivalent 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. Different 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.