In this paper we generalize constructions of non-commutati
ve integrable systems to the context of weakly Hamiltonian actions on Poisson manifolds. In particular we prove that abelian weakly Hamil
tonian actions on symplectic manifolds split into Hamiltonian and non-Hamiltonian factors, and explore generalizations in the Poisson setting.
Miranda, E., MARTINEZ, D. Weakly Hamilltonian actions. "Journal of geometry and physics", 2016.