Classification of Darboux transformations for operators of the form $\partial_x\partial_y +a \partial_x + b\partial_y +c$

Darboux transformations are non-group type symmetries of linear differential operators. One can define Darboux transformations algebraically by the intertwining relation $ML=L_1M$ or the intertwining relation $ML=L_1N$ in the cases when the first one is too restrictive. Here we show that Darboux transformations for operators of the form $\partial_x\partial_y +a \partial_x + b\partial_y +c$ (often referred to as 2D Schr\"odinger operators) are always compositions of atomic Darboux transformations of two different types. This is in contrast with the case of 1D Schr\"odinger operators and other 1D operators, where there is only one atomic kind of Darboux transformations.