the central region.

If an orbit is located in the asymptotic region where the influence of a scalar field on the spacetime geometry vanishes, then the angle of precession is always positive, as well as in the Schwarzschild spacetime. There is also the intermediate spacetime region where the scalar field is relatively weak and the precession is absent for some special values of $J$ and $E$. More exactly, for a given SFNS spacetime, the domain (in the $(J,E)$ plane) of the existence of noncircular bound orbits will be separated into two open parts, with $\Delta\varphi<0$ and $\Delta\varphi>0$, by the curve defined by the equation $\Delta\varphi(E,J)=0$.

At the end of this section, it is important to note that we have a powerful degree of freedom in the choice of the self-interaction potential (in the wide class of physically admissible potentials) of a scalar field or, equivalently, in the choice of the distribution of the scalar field itself. To see a degree of universality of the construction under consideration, one can take into account the identity $2(\xi-3m)\mathrm{e}^{\Phi}=\left[(\xi-3m)^2\right]'$ and rewrite the expression (\[A’\]) as an equation for $\xi$ (and thus for $\phi$). The resulting equation, $$\label{}
\left[(\xi-3m)^2\right]'=2rA-r^2A'
\nonumber$$ with the asymptotic condition $(\xi-3m)^2\,\rightarrow{}\,r^2-\,6mr$ 