Superintegrability of rational Ruijsenaars-Schneider systems and their action-angle duals

We explain that the action-angle duality between the rational Ruijsenaars-Schneider and hyperbolic Sutherland systems implies immediately the maximal superintegrability of these many-body systems. We also present a new direct proof of the Darboux form of the reduced symplectic structure that arises in the 'Ruijsenaars gauge' of the symplectic reduction underlying this case of action-angle duality. The same arguments apply to the BCn generalization of the pertinent dual pair, which was recently studied by Pusztai developing a method utilized in our direct calculation of the reduced symplectic structure.