  We make a detailed analysis of Newtonian as well as relativistic core-shell
models recently proposed to describe a black hole or neutron star surrounded by
shells of matter, and in a seminal sense also galaxies, supernovae and star
remnants since there are massive shell-like structures surrounding many of them
and also evidences for many galactic nuclei hiding black holes. We discuss the
unicity of the models in relation to their analyticity at the black hole
horizon and also to the full elimination of conical singularities. Secondly, we
study the role played by the presence/lack of discrete reflection symmetries
about equatorial planes in the chaotic behavior of the orbits, which is to be
contrasted with the almost universal acceptance of reflection symmetries as
default assumptions in galactic modeling. We also compare the related effects
if we change a true central black hole by a Newtonian central mass. The
numerical findings are: 1- The breakdown of the reflection symmetry about the
equatorial plane in both Newtonian and relativistic core-shell models does i)
enhance in a significant way the chaoticity of orbits in reflection symmetric
oblate shell models and ii) inhibit significantly also the occurrence of chaos
in reflection symmetric prolate shell models. In particular, in the prolate
case the lack of the reflection symmetry provides the phase space with a robust
family of regular orbits that is otherwise not found at higher energies. 2- The
relative extents of the chaotic regions in the relativistic cases (i. e. with a
true central black hole) are significantly larger than in the corresponding
Newtonian ones (which have just a $-1/r$ central potential).
