A note on the KAM theorem for partially integrable Hamiltonian systems
  We provide a symplectic reduction of a partially integrable Hamiltonian system to a completely integrable one. The KAM theorem is applied to this reduced completely integrable Hamiltonian system. Its KAM perturbation generates a perturbation of the original partially integrable system such that the set of tori kept by this perturbation is large. 
G. Giachetta, L. Mangiarotti and G. Sardanashvily