using a parameter choice $$\begin{aligned}
    \tau = t, 
\end{aligned}$$ we can give the embedding of $\Sigma$, $ {x}^\mu(\tau, \sigma)$, in the form $$\begin{aligned}
     {x}^\mu (\tau , \sigma ) = 
        \left\{\begin{array}{l}
             {t} (\tau ) = \tau \\
             {x}^k (\sigma ) 
                = ( \rho(\sigma),  \varphi(\sigma),  {z}(\sigma)). 
        \end{array}
 \right. 
 \label{ParAnsSR}\end{aligned}$$

Reduced system
--------------

We should obtain the geodesics in the three-dimensional orbit space. The action for the geodesics with respect to the metric is equivalent to the action $$\begin{aligned}
 S  &=\frac12 \int \left[ \frac{1}{N} \tilde h_{ij}  x'^i x'^j 
        +N\right]d\sigma 
\label{P_action_formal}
\\
  &= \frac{1}{2}\int \left[\frac{1}{N}\left\{(1-\Omega^2{\rho }^2 ){\rho}'^2 
        +  {\rho}^2  {\varphi}'^2 
        + (1-\Omega^2{\rho }^2){z}'^2\right\} +N\right]d\sigma, 
\label{P_action}\end{aligned}$$ where the prime denotes the derivative with respect to $\sigma$, and $N$ is an arbitrary function of $\sigma$ which is related to the reparametrization invariance of the curve.

In the action , $z$ and $\varphi$ are cyclic (ignorable) coordinates. Here, using the fact that the momentum conjugate to $z$ is conserved we pay attention to curves which is projected on $\rho-\varphi$ plane. We can interpret the projected curves in two ways; One is geodesics in a two-dimensional curved space, and the other is spatial orbits of a particle moving in a potential in two-dimensional flat space. We proceed along 