[en] The methods of classical differential geometry are used to obtain new non-linear equations, admitting the Lax representation. The procedure of the Lax representation obtaining is conducted for a sphere in threedimensional euclidean space using curvilinear system of coordinates, in which families of geodesic ellipses and hyperbolas are the coordinant lines. A new non-linear equation, which has never been discussed in literature, is obtained. It describes inher geometry of the sphere in selected system of coordinates and associated linear spectral problem in terms of (3x3) matrices is constructed for it