  The aim of this work is to continue the analysis, started in
arXiv:2105.02108, of the dynamics of a point-mass particle $P$ moving in a
galaxy with an harmonic biaxial core, in whose center sits a Keplerian
attractive center (e.g. a Black Hole). Accordingly, the plane $\mathbb R^2$ is
divided into two complementary domains, depending on whether the gravitational
effects of the galaxy's mass distribution or of the Black Hole prevail. Thus,
solutions alternate arcs of Keplerian hyperbolae with harmonic ellipses; at the
interface, the trajectory is refracted according to Snell's law. The model was
introduced in arXiv:1501.05577, in view of applications to astrodynamics. In
this paper we address the general issue of periodic and quasi-periodic orbits
and associated caustics when the domain is a perturbation of the circle, taking
advantage of KAM and Aubry-Mather theories.
