Spherical singularities in compactified Ruijsenaars--Schneider systems

We investigate certain Liouville integrable systems constructed earlier via reduction of the quasi-Hamiltonian double of $\mathrm{SU}(n)$. These systems live on compact connected symplectic manifolds of dimension $2(n-1)$ and can be interpreted as compactified trigonometric Ruijsenaars--Schneider systems. Depending on the value of a parameter $0<y< \pi$, they arise in two drastically different forms: in type (i) these are toric systems, while in the type (ii) cases they possess globally continuous action variables that generate a Hamiltonian torus action (only) on a dense open subset of the phase space. The principal goal of the paper is to study those fibers of the action map (alias the $\mathbb{T}^{n-1}$ momentum map) which are contained in the complement of the domain of the densely defined torus action occurring in the type (ii) cases. We demonstrate that all such `singular fibers' are smooth connected isotropic submanifolds. We also work out a model of the fibers as quotient spaces of certain subgroups of $\mathrm{SU}(n)$ with respect to an action of another subgroup. The general results are exemplified by determining the vertices of the polytope filled by the action variables in the simplest type (ii) cases that appear for any $n\geq 4$ with $\pi/(n-1) <y < \pi/(n-2)$, and proving that the fibers over the `singular vertices' are diffeomorphic to $S^3 \simeq \mathrm{SU}(2)$ in these cases. In this way, our findings enrich the set of examples of Liouville integrable systems with spherical singularities.