Canonical bases and collective integrable systems

Let K be a non-abelian compact connected Lie group. We show that every Hamiltonian K-manifold M admits a Hamiltonian torus action of the same complexity on a connected open dense subset U. The moment map for this torus action extends continuously to all of M. This generalizes the celebrated Gelfand-Zeitlin integrable systems introduced by Guillemin and Sternberg. In the process we develop a general framework for integrating gradient Hamiltonian vector fields on degenerations. This framework can be applied to degenerations of varieties that are possibly singular or non-compact. It recovers the results of Harada and Kaveh as a special case. Our main result on Hamiltonian K-manifolds is obtained by applying this framework to the base affine space of the complexification of K.