and Lorentzian signature in the case of spacelike Killing vector, $f>0$.

The action gives the length of ${\cal C}$ with respect to the metric $-f h_{\mu\nu}$ on the orbit space of $\xi$. Therefore, the problem for finding solutions of cohomogeneity-one string associated with $\xi$ reduces to the problem for solving three-dimensional geodesic equations with respect to the metric $-f h_{\mu\nu}$.

Equations of Motion for Stationary Rotating Strings 
----------------------------------------------------

We consider stationary rotating strings in Minkowski spacetime with the metric in the cylindrical coordinate for an inertial reference frame, $$\begin{aligned}
 ds^2 = -d\bar t^2 + d\bar\rho^2 + \bar \rho^2 d\bar \varphi^2 +d\bar z^2. 
\label{MinMetCyl} \end{aligned}$$ The world surface is tangent to the Killing vector field $$\begin{aligned}
    \xi = \partial_{\bar t} +\Omega \partial_{\bar\varphi}, 
\label{Killing}
\end{aligned}$$ where $\Omega$ is a constant denoting the angular velocity. We introduce a new coordinate $$\begin{aligned}
    x^\mu 
        =( {t},  {\rho}, {\varphi},  {z})
        =(\bar t, \bar \rho, \bar \varphi-\Omega \bar t, \bar z)
\end{aligned}$$ such that $$\begin{aligned}
    \xi = \partial_{ {t}} . 
\end{aligned}$$ The coordinate system $ {x}^\mu$ gives the rigidly rotating reference frame with the angular velocity $\Omega$. In this coordinate system, we rewrite the metric of Minkowski spacetime as $$\begin{aligned}
 ds^2 &=  {g}_{\mu \nu} d {x}^\mu d 