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Abstract. Poincare-invariant generalizations of the Galilei-invariant Calogero-Moser AΓ-particle systems are studied. A quantization of the classical integrals S 1? ...,S N is presented such that the operators S i9 ...,S N mutually commute. As a corollary it follows that S i9 ... 9 S N Poisson commute. These results hinge on functional equations satisfied by the Weierstrass σ-and 0*-functions. A generalized Cauchy identity involving the σ-function leads to an N x N matrix L whose symmetric functions are proportional to S l5 ... 9 S N .

We present a new solution method for a class of first order analytic difference equations. The method yields explicit "minimal" solutions that are essentially unique. Special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic, trigonometric and elliptic type-Euler's gamma function being of rational type. We study these generalized gamma functions in considerable detail. The scattering and weight functions (u· and w-functions) associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions. We obtain detailed information on these u-and w-functions, exploiting the difference equations they satisfy.

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Abstract. We present and study Poincare-invariant generalizations of the Galilei-invariant Toda systems. The classical nonperiodic systems are solved by means of an explicit action-angle transformation.