respect to the spacetime metric. For our purposes, it is convenient to write the metric of a spherically symmetric spacetime in the Schwarzschild-like coordinates as $$\label{metric}
    ds^2=A dt^2- \frac{\,dr^2}{f}- r^{2}(d\theta^2+\sin^{2}\!\theta\, d\varphi^2),$$ where the metric functions $A$ and $f$ depend only on the radial coordinate $r$. Writing $$A(r)=f(r)\mathrm{e}^{2\Phi(r)},$$ we obtain the Einstein-Klein-Gordon equations in the form $$\label{EKG1}
-\frac{f'}{r}-\frac{f-1}{r^2}={\phi'}^2 f + 2V\,,$$ $$\label{EKG2}
\frac{f}{r}\left(\!2\Phi'+\frac{f'}{f}\right)+\frac{f-1}{r^2}=
{\phi'}^2 f - 2V\,,$$ $$\label{EKG3}
-f\phi''-\frac{\phi'}{2}f'-\phi' f\left(\!\Phi' +
\frac{1}{2}\frac{f'}{f}+\frac{2}{r}\right)+ \frac{dV}{d\phi}=0\,,$$ where a prime denotes differentiation with respect to $r$. Now these equations can be reduced to two independent ones by summing equations (\[EKG1\]) and (\[EKG2\]) (with the result $\Phi'=r{\phi'}^2$), and then eliminating $\Phi'$ from (\[EKG3\]).

The functions $A$ and $f$, which completely determine geodesic motion in the spacetime, should be the result of solving equations (\[EKG1\])–(\[EKG3\]). However, we have no a priori knowledge of the form of $V(\phi)$ and have to study bound orbits for all physically admissible self-interaction potentials at once. Our approach is based on the following result [@Solovyev2012]:

*A general static, spherically symmetric, asymptotically flat solution of equations (\[EKG1\])–(\[EKG3\]) with an arbitrary self-interaction potential is given by the quadratures* $$\label{F-xi}
\Phi(r)=-\!\int_{r}^{\,\infty} {\phi'}^{2}rdr\,,\quad \xi(r)=r+\int_{r}^{\,\infty}
\left(1-\mathrm{e}^{\Phi}\right)\!dr\,,$$ $$\label{A-f}
A(r)=2r^{2}\!\int_{r}^{\,\infty} \frac{\,\xi-3m}{\,r^4}\,\mathrm{e}^{\Phi}dr\,, \quad f(r)=\mathrm{e}^{-2\Phi}A\,,$$ $$\label{V}
\widetilde{V}(r)=\frac{1}{2r^2}\!\left(1-3f+ r^2{\phi'}^{2}\!f+ 2\,\mathrm{e}^{-\Phi}\,\frac{\,\xi-3m}{r}\right),$$ *where the 