An algebraic method to construct the Darboux matrix

We present an effective procedure to construct the 1-soliton Darboux matrix. Our approach, based on the Zakharov–Shabat–Mikhailov’s dressing method, is especially useful in the case of non-canonical normalization and for non-isospectral linear problems. The construction is divided into two steps. First, we represent a given linear problem as a system of some algebraic constraints on two matrices. In this context we introduce and discuss invariants of the Darboux matrix. Second, we derive the Darboux matrix demanding that it preserves the algebraic constraints. In particular, we consider in details the restrictions imposed by various reduction groups on the form of the Darboux matrix.