Birkhoff normalization, bifurcations of Hamiltonian systems and the Deprits formula

We consider Hamiltonian autonomous systems with n degrees of freedom near a singular point. In the case of absence of resonances of order less than or equal to 4 we present a direct computation of the Birkhoff normal form. In the case of two degrees of freedom, we study 1-parameter deformations of the 0: 1, 1: 1 and 2: 1 resonant singularities. The obtained results are used in a direct derivation of the Deprits formula for the isoenergetic degeneracy determinant in the restricted three-body problem. © 2013 The Author(s).