Integrable systems from Poisson reductions of generalized Hamiltonian torus actions

We develop a set of sufficient conditions for guaranteeing that an integrable system with a symmetry group K on a manifold M descends to an integrable system on a dense open subset of the quotient Poisson space $M/K$. The higher dimensional phase space M carries a bivector PM yielding a bracket on C^∞(M) such that C^∞(M)K is a Poisson algebra. The unreduced system on M is supposed to possess `action variables' that generate a proper, effective action of a group of the form U(1)₁ × R₂ and descend to action variables of the reduced system. In view of the form of the group and since PM could be a quasi-Poisson bivector, we say that we work with a generalized Hamiltonian torus action. The reduced systems are in general superintegrable owing to the large set of invariants of the proper Hamiltonian action of U(1)₁ × R₂. We present several examples and apply our construction for solving open problems regarding the integrability of systems obtained previously by reductions of master systems on doubles of compact Lie groups: the cotangent bundle, the Heisenberg double and the quasi-Poisson double. Furthermore, we offer numerous applications to integrable systems living on moduli spaces of flat connections, using the quasi-Poisson approach.