Differential Transformations of Parabolic Second-Order Operators in the Plane

The theory of transformations for hyperbolic second-order equations in the plane, developed by Darboux, Laplace and Moutard, has many applications in classical differential geometry [12, 13], and beyond it in the theory of integrable systems [14, 19]. These results, which were obtained for the linear case, can be applied to nonlinear Darboux-integrable equations [2, 7, 15, 16]. In the last decade, numerous generalizations of the classical theory have been developed. Among them there are generalizations to the case of systems of hyperbolic equations in the plane [3, 5, 6, 22], and generalizations to the case of hyperbolic equations with more than