  Birkhoff normal form is a power series expansion associated with the local
behavior of the Hamiltonian systems near a critical point. It is known to be
convergent for integrable systems under some non-degeneracy conditions. By
means of an expression of the inverse of Birkhoff normal form by a period
integral, analytic continuation of the Birkhoff normal forms is considered for
the free rigid body dynamics on SO(3). The global behavior of the Birkhoff
normal forms is clarified in relation to the monodromy of an elliptic fibration
which naturally arises from the free rigid body dynamics.
