only similar orbit, having the same motionless pericentre and apocentre as the orbit around the SFNS, has the angle of precession $\Delta\varphi=2\pi$. The middle and right panels of Fig. \[fig3\] represent two very highly elongated orbits around the SFNS spacetime with $a=50$ and the Schwarzschild black hole, respectively. These orbits have approximately the same apocentre radius, which is larger than the size of the central part of the orbits by a factor of about 50. From the point of view of a distant observer, the orbits can look very similar or even being observationally indistinguishable from one another, whereas their angles of precession are, respectively, $-3\pi/2$ and $+\pi/2$. The insets in the middle and right panels show the obvious difference in the behaviours of the orbits in the central regions; this example explains why we need the observations of both pericentre and apocentre of elongated orbits.

Conclusions
===========

In this paper, we considered a model of a spherically symmetric strongly gravitating massive object surrounded by a self-gravitating nonlinear scalar field, having in mind the centre of a galaxy surrounded by dark matter. This idealized model is treated in a fully analytical manner and, in this way, we have found some 