Solution of Gauss--Codazzi Equation with Applications in the Tzitzeica Equation

The surface in R³ associated with the Tzitzeica equation is considered. By curvature coordinate ransformation and surface imbedding, the Gauss--Codazzi equation is presented. Resorting to the solutions of the Gauss--Codazzi equation, the solution of the Tzitzeica equation is obtained under a restrictive condition.