Intertwine operators and elementary darboux transforms in differential rings and modules

This paper presents an algebraic structure allowing to construct an elementary Darboux transformation. The set of necessary propositions is listed and the generic Darboux theorem is formulated for differential rings. The result is given for the generalized Zakharov-Shabat problem. The conditions for the appearance of the Schlesinger transformation and a construction of an analogue of supersymmetry algebra are shown starting from the intertwine relation. Some applications in the theory of nonlinear equations are also discussed.