The global flow of the hyperbolic restricted three-body problem

Two mass points of equal masses m ₁ = m ₂ > 0 move under Newton's law of attraction in a non-collision hyperbolic orbit while their center of mass is at rest. We consider a third mass point, of mass m ₃ = 0, moving on the straight line L perpendicular to the plane of motion of the first two mass points and passing through their center of mass. Since m ₃ = 0, the motion of m ₁ and m ₂ is not affected by the third, and from the symmetry of the motion it is clear that m ₃ remains on the line L. The hyperbolic restricted 3-body problem is to describe the moton of m ₃. Our main result is the characterization of the global flow of this problem.