ELLIPTIC SOLITONS, FUCHSIAN EQUATIONS, AND ALGORITHMS

It is shown how the elliptic finite-gap potentials of the Schrodinger equation give rise to a family of solvable linear differential equations of the Fuchs class on the plane and on the torus: the latter case cannot be integrated via realizations of the Zinger Kovacic type algorithms known in the Picard-Vessiot theory. For the arising Fuchsian equations, monodromy groups and their representations are constructed, the differential Galois group is described, together with a (recursive) method for calculation of the objects involved therein.