On the Birkhoff normal form of a completely integrable hamiltonian system near a fixed point with resonance

We consider an integrable Hamiltonian system with a real analytic Hamiltonian H near an elliptic fixed point P. If H has a simple resonance and admits a semisimple Hessian at P we show that there exists a real analytic change of coordinates which brings the Hamiltonian into normal form. In the new coordinates, the level sets of the system are analyzed in terms of the nature of the simple resonance.