\right) - I a \Phi, 
  \quad P_{\theta} = I (a {\cal{E}} - \Phi),                            
  \quad K = I^2 (a {\cal{E}} - \Phi)^2.                         \label{e18}\end{aligned}$$ The proper time of the particle $\tau$ is related to the affine parameter $\lambda$ by $\tau = m \lambda$. The constants of motion are: energy ($\cal{E}$), related to the stationarity of the geometry, axial angular momentum ($\Phi$), related to the axial symmetry of the geometry, ‘total’ angular momentum ($K$), related to the hidden symmetry of the geometry. For the equatorial motion, $K$ is restricted through Eq. (\[e18\]) following from the conditions on the latitudinal motion.[@Stu:1983:BULAI:] Notice that $\cal{E}$ and $\Phi$ cannot be interpreted as energy and axial angular momentum at infinity, since the spacetime is not asymptotically flat.

The equatorial motion is governed by the constants of motion ${\cal{E}},\ \Phi$. Its properties can be conveniently determined by an “effective potential” given by the condition $R(r) = 0$ for turning points of the radial motion. It is useful to define specific energy and specific angular momentum by the relations $$\label{e21}
        E \equiv \frac{I {\cal{E}}}{m}, \ L \equiv \frac{I \Phi}{m}.$$ Solving the equation $R(r) = 0$, we find the effective potential in the form $$\begin{aligned}
        E_{(\pm)}(r;L,a,y) &\equiv&
\left[\left(1+y a^2 \right)r\left(r^2 