Invariant tori of rectilinear type in the spatial three-body problem

In the context of the spatial three-body problem and KAM theory, specifically in the regime where the Hamiltonian function is split as the sum of two Keplerian terms plus a small perturbation, we deal with quasi-periodic motions of the three bodies such that two of the three particles (the so-called inner bodies) describe near-collision orbits. More precisely the inner bodies never collide, but they follow orbits that are bounded straight lines or close to straight lines. The motion of the inner bodies occurs either near the axis that is perpendicular to the invariable plane (i.e. the fixed plane orthogonal to the angular momentum vector that passes through the centre of mass of the system) or near the invariable plane. The outer particle's trajectory has an eccentricity varying between zero and a value that is upper bounded by e2M<1 and lies near the invariable plane. The three bodies' orbits fill in invariant 5-tori and when the inner particles move in an axis perpendicular to the invariable plane, they correspond to new solutions of the three-body problem. Our approach consists in a combination of a regularisation procedure with the construction of various reduced spaces and the explicit determination of sets of symplectic coordinates. The various reduced spaces we build depend on what symmetries are taken into consideration for the reduction. Moreover we apply an iso-energetic theorem by Han, Li and Yi on the persistence of quasi-periodic solutions for Hamiltonian systems with high-order proper degeneracy. All these elements allow us to calculate explicitly the torsions for the possible combinations that the three particles' motions can achieve.