Semi-global symplectic invariants of the Euler top

We compute the semi-global symplectic invariants near the hyperbolic equilibrium points of the Euler top. The Birkhoff normal form at the hyperbolic point is computed using Lie series. The actions near the hyperbolic point are found using Frobenius expansion of its Picard-Fuchs equation. We show that the Birkhoff normal form can also be found by inverting the regular solution of the Picard-Fuchs equation. Composition of the singular action integral with the Birkhoff normal form gives the semi-global symplectic invariant. Finally, we discuss the convergence of these invariants and show that in a neighbourhood of the separatrix the pendulum is not symplectically equivalent to any Euler top.