The Kepler Problem with Anisotropic Perturbations

We study a 2-body problem given by the sum of the Newtonian potential and an anisotropic perturbation that is a homogeneous function of degree $-\beta$, $\beta \ge 2$. For $\beta >2$, the sets of initial conditions leading to collisions/ejections and the one leading to escapes/captures have positive measure. For $\beta >2$ and $\beta \mathrm{\ne }3$, the flow on the zero-energy manifold is chaotic. For $\beta =2$, a case we prove integrable, the infinity manifold of the zero-energy level is a disconnected set, which has heteroclinic connections with the collision manifold.