Dynamics close to a non semi-simple 1:-1 resonant periodic orbit

In this work, our target is to analyze the dynamics around the $1:-1$ resonance which appears when a family of periodic orbits of a real analytic three-degree of freedom Hamiltonian system changes its stability from elliptic to a complex hyperbolic saddle passing through degenerate elliptic. Our analytical approach consists of computing, up to some given arbitrary order, the normal form around that resonant (or critical) periodic orbit.
Hence, dealing with the normal form itself and the differential equations related to it, we derive the generic existence of a two-parameter family of invariant 2D tori which bifurcate from the critical periodic orbit. Moreover, the coefficient of the normal form that determines the stability of the bifurcated tori is identified. This allows us to show the Hopf-like character of the unfolding: elliptic tori unfold "around'' hyperbolic periodic orbits (case of direct bifurcation) while normal hyperbolic tori appear "around'' elliptic periodic orbits (case of inverse bifurcation). Further, the parametrization of the main invariant objects as well as a global description of the dynamics of the normal form are also given.