  We analyze, through a geometric description, the sequence of bifurcations of
periodic orbits in a Hamiltonian model derived from the normalization of the
secular 3D planetary three body problem. Stemming from the results in
(Mastroianni & Efthymiopoulos 2023) we analyze the phase space of the
corresponding integrable approximation. In particular, we propose a normal form
leading to an integrable Hamiltonian whose sequence of bifurcations is
qualitatively the same as that in the complete system. Using as representation
of the phase space the 3D-sphere in the Hopf variables space, we geometrically
analyze phase-space dynamics through the sequence of bifurcations leading to
the appearance of fixed points of the secular Hamiltonian flow, i.e., periodic
orbits in the complete system. Moreover, through a semi-analytical method, we
find the critical values of the second integral giving rise to pitchfork and
saddle-node bifurcations characterising the dynamics.
