Poisson-Lie analogues of spin Sutherland models revisited

Some generalizations of spin Sutherland models descend from `master integrable systems' living on Heisenberg doubles of compact semisimple Lie groups. The master systems represent Poisson--Lie counterparts of the systems of free motion modeled on the respective cotangent bundles and their reduction relies on taking quotient with respect to a suitable conjugation action of the compact Lie group. We present an enhanced exposition of the reductions and prove rigorously for the first time that the reduced systems possess the property of degenerate integrability on the dense open subset of the Poisson quotient space corresponding to the principal orbit type for the pertinent group action. After restriction to a smaller dense open subset, degenerate integrability on the generic symplectic leaves is demonstrated as well. The paper also contains a novel description of the reduced Poisson structure and a careful elaboration of the scaling limit whereby our reduced systems turn into the spin Sutherland models.