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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.

We present an explicit construction of an action-angle map for the nonrelativistic N-particle Sutherland system and for two different generalizations thereof, one of which may be viewed as a relativistic version. We use the map to obtain detailed information concerning dynamical issues such as oscillation periods and equilibria, and to obtain simple formulas for partition functions. The nonrelativistic and relativistic Sutherland systems give rise to dual integrable systems with a solitonic long-time asymptotics that is explicitly described. We show that the second generalization is self-dual, and that its reduced phase space can be densely embedded in P N~l with its standard Kahler form, yielding commuting global flows. In a certain limit the reduced action-angle map converges to the quotient of Fourier transformation on C^ under the standard projection

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We survey results on Galilei-and Poincare-invariant CalogeroMoser and Toda N-particle systems, both in the context of classical mc hanics and of quantum mechanics. Special attention is given to integrability issues and interconnections between the various models. Action-angle and joint eigenfunction transforms are also considered, and some novel results on N = 2 eigenfunctions of hyperbolic Askey-Wilson type and of relativistic elliptic type are sketched.