ABSTRACT. Combining algebro-geometric methods and factorization techni-
ques for finite difference expressions we provide a complete and self-contained
treatment of all real-valued quasi-periodic finite-gap solutions of both the Toda
and Kac-van Moerbeke hierarchies.
In order to obtain our principal new result, the algebro-geometric finite-
gap solutions of the Kac-van Moerbeke hierarchy, we employ particular com-
mutation methods in connection with Miura-type transformations which ena-
ble us to transfer whole classes of solutions (such as finite-gap solutions) from
the Toda hierarchy to its modified counterpart, the Kac-van Moerbeke hierar-
chy, and vice versa.
Keywords and phrases.
Jacobi operators, Toda hierarchy, Kac-van Moerbeke hierarchy