A theorem gives sufficient conditions for an integrable system to descend through Poisson reduction, with explicit generalized action variables, applied to moduli spaces of flat connections and to the three doubles of compact Lie groups.
Non-commutative Integrability, Moment Map and Geodesic Flows
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abstract
The purpose of this paper is to discuss the relationship between commutative and non-commutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the bi-invariant metric on any bi-quotient of a compact Lie group is integrable in non-commutative sense by means of polynomial integrals, and therefore, in classical commutative sense by means of $C^\infty$--smooth integrals.
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Integrable systems from Poisson reductions of generalized Hamiltonian torus actions
A theorem gives sufficient conditions for an integrable system to descend through Poisson reduction, with explicit generalized action variables, applied to moduli spaces of flat connections and to the three doubles of compact Lie groups.