  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.
