\left(\frac{2\sqrt[4]{20}\sqrt{r(r+a)}} {\big(\sqrt{5}+2\big)r+\sqrt{5}\,a\,}\right),
\nonumber$$ and $F(\lambda,k)$ denotes the incomplete elliptic integral of the first kind in the Legendre normal form, $$F(\lambda,k)=\int_{0}^{\,\sin\lambda} \frac{dt}{\,\sqrt{1-t^2}\sqrt{1-k^2t^2}\,}.
\nonumber$$ One has $\phi=\sqrt[4]{4/5}\,\,F(\lambda_0,k)\,-\, \sqrt{16r/5a}\,+\,O(r^{3/2})$ near $r=0$ ($\phi(0)\approx2.498$) and $\phi=2a/r+O(r^{-2})$ at infinity. As a function of $r$, the potential can be obtained directly from equation (\[V\]) but has a cumbersome form; (\[V\]) and (\[phi-F\]) determine, parametrically, $V(\phi)$. This potential, which has arisen in a simple demo example, is not physically interesting because it is negative everywhere in the interval $(0,\infty)$. However, there exist solutions whose self-interaction potential is positive in the region $r>r_p$ with $r_p$ sufficiently close to the centre [@Nikonov2008].

![image](Fig1a){width="31.30000%"} ![image](Fig1b){width="32.00000%"} ![image](Fig1c){width="31.90000%"}

![image](Fig2a){width="31.80000%"} ![image](Fig2b){width="31.80000%"} ![image](Fig2c){width="31.80000%"}

![image](Fig3a){width="30.80000%"} ![image](Fig3b){width="32.60000%"} ![image](Fig3c){width="32.60000%"}

Our main goal (of considerable observational interest) is to compare orbits which have the same pericentre and apocentre radii and a comparable number of oscillations, but which either are in different spacetimes or have different parameters. Fig. \[fig1\] shows the typical behaviour of the function $\xi(r)$ belonging to the family (\[xi\]), the metric function $A(r)$ of a SFNS together (and in comparison) with the corresponding metric function of the Schwarzschild solution of the same mass, and a typical bound orbit in the latter spacetime. It is important to note 