Differential Galois theory and Darboux transformations for Integrable Systems

We apply the Differential Galois Theory of linear partial differential systems to the Bäcklund-Darboux transformations of the AKNS solitonic partial differential equations. We prove that the Galois group of the transformed system is isomorphic to a subgroup of the Galois group of the initial system. As an example, we study the integrability in closed form of the linear systems corresponding to the solitonic solutions of KdV equation.