Analytic extension of the Birkhoff normal forms for the free rigid body dynamics on SO(3)

Birkhoff normal form is a power series expansion associated with the local behaviour of the Hamiltonian systems near a critical point. It is known that around the critical point one can take a convergent canonical transformation which puts the Hamiltonian into Birkhoff normal form for integrable systems under some non-degeneracy conditions. By means of an expression of the derivative for 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). It is shown that the monodromy of the analytic continuation for the derivative of the inverse for the Birkhoff normal forms coincides with that of an elliptic fibration which naturally arises from the dynamics.