Classical and quantum-mechanical systems of Toda lattice type. I

The structure of the commutant of Laplace operators in the enveloping and Poisson algebra of certain generalized ax +b groups leads (in this article) to a determination of classical and quantum mechanical first integrals to generalized periodic and non-periodic Toda lattices. Certain new Hamiltonian systems of Toda lattice type are also shown to fit in this framework. Finite dimensional Lax forms for the (periodic) Toda lattices are given generalizing results of Flaschke.Research partially supported by NSF grant MCS 79-03223Research partially supported by NSF grant MCS 79-03153