Bifurcations of the Hill's region in the three body problem

In the spatial three body problem, the topology of the integral manifolds $\mathfrak{M}(c,h)$ (i.e. the level sets of energy $h$ and angular momentum $c$, as well as center of mass and linear momentum) and the Hill's regions $\mathfrak{H}(c,h)$ (the projection of the integral manifold onto position coordinates) depends only on the quantity $\nu = h|c|^2.$ It was established by Albouy and McCord-Meyer-Wang that, for $h < 0$ and $c \neq 0$, there are exactly eight bifurcation values for $\nu$ at which the topology of the integral manifold changes. It was also shown that for each of these values, the topology of the Hill's region changes as well. In this work, it is shown that there are no other values of $\nu$ for which the topology of the Hill's region changes. That is, a bifurcation of the Hill's region occurs if and only if a bifurcation of the integral manifold occurs.