Laplace transformations as the only degenerate Darboux transformations of first order

@article{Shemyakova2012LaplaceTA,
title={Laplace transformations as the only degenerate Darboux transformations of first order},
author={Ekaterina Shemyakova},
journal={Programming and Computer Software},
year={2012},
volume={38},
pages={105-108}
}

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.