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We explain that the action-angle duality between the rationalRuijsenaars-Schneider and hyperbolic Sutherland systems implies immediately themaximal superintegrability of these many-body systems. We also present a newdirect proof of the Darboux form of the reduced symplectic structure thatarises in the `Ruijsenaars gauge' of the symplectic reduction underlying thiscase of action-angle duality. The same arguments apply to the BC(n)generalization of the pertinent dual pair, which was recently studied byPusztai developing a method utilized in our direct calculation of the reducedsymplectic structure.

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