dark matter is mainly concentrated around them. Another problem is that the nature of dark matter and its distribution near galactic centres remain unknown at present. This means that a meaningful interpretation of the observations should be based on an appropriate mathematical model of the central regions.

In this paper, we model dark matter by a nonlinear scalar field which is assumed to be minimally coupled to gravity. This model is an interesting alternative to the cold dark matter phenomenology (see e.g. [@Lee1996; @Matos2004; @Robles2012; @Benisty2017; @Bernal2019] and references therein). In the centres of galaxies, the distribution of dark matter seems to be spherically symmetric. Our aim is to study bound orbits of test particles near the centres of static, spherically symmetric configurations of a self-gravitating nonlinear scalar field; in our terminology, a bound orbit is a bounded *complete* geodesic in an asymptotically flat spacetime. The motivation for the choice of this special and idealized model is to treat the problem in a fully analytical and self-consistent manner. For the orbital motion of a test particle near a compact hairy object, this model allows us to find out some important features, which can possibly be observed in real astrophysical systems, 