On the existence of Darboux transformations for banded matrices

The Darboux transformations provide, joint with other applications, a method for obtaining solutions of some integrable systems. In this work, the concepts of Darboux factorization and Darboux transformations for arbitrary Hessenberg banded matrices are analyzed. Specifically, the existence of this kind of factorization is studied, and some sufficient conditions for the uniqueness are determined.