  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.
