Monodromy of the matrix Schrodinger equations and Darboux transformations

A Schrodinger operator L = -d(2)/dz(2) + U(z) with a matrix-valued rational potential U(z) is said to have trivial monodromy if all the solutions of the corresponding Schrodinger equations L psi = lambda psi are single-valued in the complex plane z is an element of C for any lambda. A local criterion of this property in terms of the Laurent coefficients of the potential U near its singularities, which are assumed to be regular, is found. It is proved that any such operator with a potential vanishing at infinity can be obtained by a matrix analogue of the Darboux transformation from the Schrodinger operator L-0 = -d(2)/dz(2). This generalizes the well known Duistermaat-Grunbaum result to the matrix case and gives the explicit description of the Schrodinger operators with trivial monodromy in this case.