  In this paper, we consider a Newtonian system whose relativistic counterpart
describes a superimposed halo with a black hole. Our aim is to determine how
the quadrupole and octupole moments affect the nature of the motion of a test
particle, moving in the close vicinity of the black hole. The different types
of trajectories for the test particle are mainly classified as bounded,
collisional, and escaping, by using modern color-coded basin diagrams.
Moreover, an additional analysis is carried out for distinguishing between the
different types of bounded motion (regular, sticky, and chaotic). Our results
strongly indicate that the multipole moments, along with the total orbital
energy, highly affect the final state of the test particle, while at the same
time the basin geometry of the phase space tends to be highly dominated by
collision and escape orbits.
