$N_{(2)}$ and $ \varphi$ we have $$\begin{aligned}
        &(1-q^2-\Omega^2 \rho^2)
        \left(  \rho'^2 
        +\frac{ \rho^2}{1-\Omega^2 \rho^2}
         \varphi'^2\right) =N_{(2)}^2, 
\label{2dim-norm}\\
        &\frac{(1-q^2-\Omega^2 \rho^2) \rho^2}
                {1-\Omega^2 \rho^2} \varphi'=N_{(2)}\frac{\ell}{|\Omega|}, 
\label{2dim-ell}\end{aligned}$$ where $\ell$ is a constant.

Eliminating $\varphi'$ from and we have $$\begin{aligned}
     {\rho}'^2 = \frac{N_{(2)}^2}{1-q^2-\Omega^2 \rho^2} 
        \left[1-\frac{\ell^2(1-\Omega^2 \rho^2) }
        {(1-q^2-\Omega^2 \rho^2)\Omega^2 {\rho}^2} 
        \right].
\label{integral}\end{aligned}$$ We see that $ \rho$ is bounded as $\rho_{min}\leq  \rho \leq  \rho_{max}$ because the right hand side of should be positive, where $$\begin{aligned}
    & \rho_{min}^2 = \frac{1}{2\Omega^2}\left(
            1+\ell^2-q^2 
    - \sqrt{(1+\ell +q)(1+\ell -q)(1-\ell +q)(1-\ell -q)}\right), 
\\
    & \rho_{max}^2 = \frac{1}{2\Omega^2}\left(
            1+\ell^2-q^2
    + \sqrt{(1+\ell +q)(1+\ell -q)(1-\ell +q)(1-\ell -q)}\right),
\label{minmax}
\end{aligned}$$ with $$\begin{aligned}
 &1+\ell^2 -q^2 \ge 0, \\
 &(1+\ell +q)(1+\ell -q)(1-\ell +q)(1-\ell -q) \geq 0. 
\end{aligned}$$ Allowed region of the parameters are shown in Fig.1.

If $q\neq 0$, it is easy to show that $$\begin{aligned}
    (1-\Omega^2 \rho_{min}^2)(1-\Omega^2 \rho_{max}^2) > 0.
\end{aligned}$$ It means that there are two cases: $$\begin{aligned}
    \mbox{(i)}&\quad (1-\Omega^2 \rho_{min}^2)>0 \quad 
        \mbox{and} \quad (1-\Omega^2 \rho_{max}^2) > 0 \quad 
&\mbox{for} &\quad 1-\ell^2 +q^2 > 0, \\
    \mbox{(ii)}&\quad (1-\Omega^2 \rho_{min}^2)<0 \quad 
        \mbox{and} \quad (1-\Omega^2 \rho_{max}^2) < 0 \quad 
&\mbox{for} &\quad 1-\ell^2 +q^2 < 0 .
\end{aligned}$$ In the first case, the Killing vector field $\xi$ is timelike everywhere on the world surface $\Sigma$, while $\xi$ 