Laplace transformations as the only degenerate Darboux transformations of first order

The paper is devoted to the Darboux transformations, an effective algorithm for finding analytical solutions of partial differential equations. It is proved that Wronskian-like formulas suggested by G. Darboux for the second-order linear operators on the plane describe all possible differential transformations with M of the form D _x + m(x, y) and D _y + m(x, y), except for the Laplace transformations.