Besides the main result, the paper contains an algebraic definition of a Darboux transformations for an arbitrary operator, which naturally implies additional properties that are usually had to be added artificially.
Darboux transformations for factorizable operator has lot of structure. In particular, there is a correspondence between Darboux transformations of factorizable second-order operators and Darboux transformation of first order operators. However, the correspondence is not one-to-one, and lifting back process is not immediate.
Here we find a criterion for an arbitrary operator on the plane to have an invertible Darboux transformation.
We also study Wronkian-type formulas which allow to construct Darboux transformations for many different types of operators. These are non-invertible by construction. From time to time new works appear showing that Wronskian-type formulas generate Darboux transformations for some new type of operators. However, up until this work there has been no explicit example of an operator for which Wronkiant-type formulas do NOT work. We give an explicit example in this paper, and find sufficient conditions for Wronkians to work.
The conjecture was reduced to the solution of a complicated system of non-linear PDEs. Using moving frames method (of differential geometry, due to P.Olver et al.) I have found a generating set of joint differential invariants for the involved operators. Then after re-writing in terms of invariants, the initial system split into two much simpler systems, one of which was a system of linear PDEs.