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Birkhoff normal form is a power series expansion associated with the localbehavior of the Hamiltonian systems near a critical point. It is known to beconvergent for integrable systems under some non-degeneracy conditions. Bymeans of an expression of the inverse of Birkhoff normal form by a periodintegral, analytic continuation of the Birkhoff normal forms is considered forthe free rigid body dynamics on SO(3). The global behavior of the Birkhoffnormal forms is clarified in relation to the monodromy of an elliptic fibrationwhich naturally arises from the free rigid body dynamics.

Analytic Extension of the Birkhoff Normal Forms for the Free Rigid Body Dynamics on SO(3)