Description
The compactified Ruijsenaars--Schneider systems are completely integrable Hamiltonian systems on certain smooth moduli spaces of flat SU(n) connections on the one-holed torus. The monodromy along the hole is fixed to the smallest non-trivial conjugacy class of SU(n), which can be labelled by a parameter 0<x<1. Depending on the value of this parameter, the systems 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 of dimension 2(n-1). After a review of the background material, we report our recent study of those fibers of the action
map (alias the momentum map) that are contained in the complement of the domain of the densely defined torus action occurring in the type (ii) cases. We have shown that these `singular fibers' are smooth connected isotropic submanifolds, diffeomorphic to the 3-dimensional sphere in the simplest cases, similarly to the fibers of the classical Gelfand-Cetlin systems. Based mostly on joint work with Holger Dullin, https://arxiv.org/abs/2604.18023