  We study local normal forms for completely integrable systems on Poisson
manifolds in the presence of additional symmetries. The symmetries that we
consider are encoded in actions of compact Lie groups. The existence of
Weinstein's splitting theorem for the integrable system is also studied giving
some examples in which such a splitting does not exist, i.e. when the
integrable system is not, locally, a product of an integrable system on the
symplectic leaf and an integrable system on a transversal. The problem of
splitting for integrable systems with additional symmetries is also considered.
