  Some years ago Ruijsenaars and Schneider initiated the study of mechanical
systems exhibiting an action of the Poincare algebra. The systems they
discovered were far richer: their models were actually integrable and possessed
a natural quantum version. We follow this early work finding and classifying
mechanical systems with such an action. New solutions are found together with a
new class of models exhibiting an action of the Galilean algebra. These are
related to the functional identities underlying the various Hirzebruch genera.
The quantum mechanics is also discussed.
