  In the framework of Lie transform and the global method of averaging, the
normal forms of a multidimensional slow-fast Hamiltonian system are studied in
the case when the flow of the unperturbed (fast) system is periodic and the
induced S^1-action is not necessarily free and trivial. An intrinsic splitting
of the second term in a S^1-invariant normal form of first order is derived in
terms of the Hannay-Berry connection associated with the periodic flow.
