On the connections between Ruijsenaars-Schneider systems and other areas

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?

the integrable systems in question are complexified, so for an n-body model (say Ruijsenaars-Schneider) the phase space is $$4n$$ real ($$2n$$ complex)-dimensional. If the coordinates and conjugate momenta are complex this covers trigonometric and hyperbolic at once.