with $p^{(\phi)}>0,\ (L>0)$, we call co-rotating, and the circular orbits with $p^{(\phi)}<0,\ (L<0)$ we call counter-rotating, in agreement with the case of asymptotically flat Kerr spacetimes.

The circular geodesics can be astrophysically relevant, if they are stable with respect to radial perturbations. The loci of the stable circular orbits are given by the condition $$\frac{\d^2 R}{\d r^2} \geq 0
                                     \label{e56}$$ that has to be satisfied simultaneously with the conditions $R(r) =0$ and $\d R/\d r = 0$ determining the circular orbits. The radii of the stable orbits of both families are restricted by the condition[@Stu-Sla:2004:PHYSR4:] $$r \left[6 -r +r^3 (4r -15)y \right] \mp
         8a \left[r\left(1 -yr^3 \right)^3\right]^{1/2} + 
                 a^2 \left[3 +r^2y\left(1 -4yr^3 \right)\right] \geq 0. 
                                                \label{e57}$$ The marginally stable orbits of both families are described by the relation $$\begin{aligned}
  a^2 &=& a^2_{\mathrm{ms(1,2)}}(r;y) \equiv
    \left[3 +r^2y \left(1-4yr^3\right)\right]^{-2}r
    \bigg\{\left[r -6 -r^3 (4r -15)y \right]\nonumber \\
    &\times&
      \left[3 +r^2y \left(1 -4yr^3 \right)\right]
      +32 \left(1 -yr^3 \right)^3 \pm 8 \left(1-yr^3\right)^{3/2}
      \left(1 -4yr^3 \right)^{1/2}\nonumber \\
    &\times& \left.\left\{r \left[3 -ry \left(6 +10r -15yr^3\right) \right]
      -2 \right \}^{1/2}\right \}.    \label{e58}\end{aligned}$$ The ($\pm$) sign in Eq. (\[e58\]) is not directly related to the plus-family and the minus-family orbits. The function $a^2_{\mathrm{ms(1)}}$, corresponding to the $(+)$ sign in Eq. 