  We consider three 'classical doubles' of any semisimple, connected and simply
connected compact Lie group $G$: the cotangent bundle, the Heisenberg double
and the internally fused quasi-Poisson double. On each double we identify a
pair of 'master integrable systems' and investigate their Poisson reductions.
In the simplest cotangent bundle case, the reduction is defined by taking
quotient by the cotangent lift of the conjugation action of $G$ on itself, and
this naturally generalizes to the other two doubles. In each case, we derive
explicit formulas for the reduced Poisson structure and equations of motion and
find that they are associated with well known classical dynamical $r$-matrices.
Our principal result is that we provide a unified treatment of a large family
of reduced systems, which contains new models as well as examples of spin
Sutherland and Ruijsenaars--Schneider models that were studied previously. We
argue that on generic symplectic leaves of the Poisson quotients the reduced
systems are integrable in the degenerate sense, although further work is
required to prove this rigorously.
