Coupling symmetries with Poisson structures (Q356529)

This paper considers Poisson manifolds with additional symmetries, encoded in actions of compact Lie groups. The authors first obtain an equivariant Caratheodory-Jacobi-Lie theorem for Poisson structures. Then, the existence of Weinstein's splitting theorem for the integrable system is studied. Several counterexamples are given. Moreover, the obstruction of integrability is described in terms of foliated Poisson cohomology. Finally, the authors study equivariant normal forms for regular completely integrable systems which are split.