ALGEBRO-GEOMETRIC QUASI-PERIODIC FINITE-GAP SOLUTIONS OF THE TODA ANDKAC VAN MOERBEKE HIERARCHIES

Combining algebro-geometric methods and factorization techniques for f inite difference expressions we provide a complete and self-contained treatment of all real-valued quasi-periodic finite-gap solutions of bo th the Toda and Kac-van Moerbeke hierarchies. In order to obtain our p rincipal new result, the algebro-geometric finite-gap solutions of the Kac-van Moerbeke hierarchy, we employ particular commutation methods in connection with Miura-type transformations which enable us to trans fer whole classes of solutions (such as finite-gap solutions) from the Toda hierarchy to its modified counterpart, the Kac-van Moerbeke hier archy, and vice versa.