On some integrable cases in surface theory

It is shown how to reformulate the Gauss–Codazzi system for a surface with arbitrary Gaussian curvature in the form of one second-order differential equation. A similar reformulation is performed for a surface with fixed mean curvature. In the cases of two-dimensional Bianchi surfaces of positive curvature, these equations correspond to the unitary reduction of the coupled Ernst system of the equations of general gravity.
The theta-functional description of the corresponding geometric objects is given. Bibl. 22 titles.