A superintegrable system on a symplectic manifold \((M_{2n}, \omega)\) consists of a Poisson subalgebra \(C_J(M_{2n})\) of the Poisson algebra \(C(M_{2n})\) that has rank \(2n-k\) with a Poisson center \(C_I(M)\) of rank \(k\). The Hamiltonian dynamics generated by a function \(H \in C(M)\) is called superintegrable if \(H \in C_I(M)\).
The main purpose of this paper is to prove superintegrability of Hamiltonian systems generated by functions on the coset space \(K\backslash G/K\), restricted to a symplectic leaf of the Poisson variety \(G/K\) where \(G\) is a simple Lie group with standard Poisson Lie structure and \(K\) is its subgroup of fixed points with respect to the Cartan involution.
The authors consider this paper as a sequel to [G. Schrader, Int. Math. Res. Not. 2016, No. 1, 1–23 (2016; Zbl 1346.37051); N. Reshetikhin, Commun. Math. Phys. 242, No. 1–2, 1–29 (2003; Zbl 1078.37043)].
Here they look at the superintegrability of reflecting integrable systems that arise from Cartan involutions or finite-dimensional split real Lie groups with their standard Lie-Poisson structure. In the second of the two cited papers the author proved that a choice of an appropriate low-dimensional symplectic leaf in \(\mathrm{SL}_{n+1} (\mathbb{R}/K)\) allows one to give an alternate realization of the relativistic Toda chain as a system whose phase space consists of certain symmetric tridiagonal matrices. Here the authors generalize that result to higher-dimensional leaves. They prove that for generic symplectic leaves of \(G/ K\) the Hamiltonians generated by functions on \(K\backslash G/K\) are superintegrable.