Notes on the degenerate integrability of reduced systems obtained from the master systems of free motion on cotangent bundles of compact Lie groups

The reduction of the `master system’ of free motion on the cotangent bundle $T^{\ast }G$ of a compact, connected and simply connected, semisimple Lie group is considered using the conjugation action of $G$. It is proved that the restriction of the reduced system to the smooth component of the quotient space $T^{\ast }G\mathrm{/}G$, given by the principal orbit type, inherits the degenerate integrability of the master system. The proof can be generalized easily to other interesting examples of Hamiltonian reduction.