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 $BC_n$ 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.
Viktor Ayadi. László Fehér. Tamás F. Görbe. "Superintegrability of Rational Ruijsenaars-Schneider Systems and Their Action-Angle Duals." J. Geom. Symmetry Phys. 27 27 - 44, 2012. https://doi.org/10.7546/jgsp-27-2012-27-44