I found on the literature plenty of articles dealing with connections between rational/trigonometric/elliptic Calogero-Moser systems and their relativistic generalizations (Ruijsenaars-Schneider), and other fields on mathematics and physics. There was also a topic discussing with the trichotomy R/T/E here: Groups, quantum groups and (fill in the blank).

I have a really naive question: why the hyperbolic systems do not appear in the story?