the scalar curvature is positive everywhere, while the curvature becomes negative near $ \rho=0$ when $q^2<1/3$. When $q$ is small enough, the scalar curvature has the minimum point near $ \rho=\rho_q$. Fig.\[curvature\] shows the scalar curvature as the function of $|\Omega|\rho$.

![Scalar curvature of the reduced two-dimensional space. []{data-label="curvature"}](2Dcurvature.eps){width="7cm"}

Solutions for Stationary Rotating Strings
=========================================

If we choose the function $N_{(2)}$ as $$\begin{aligned}
    N_{(2)}=1-q^2-\Omega^2\rho^2, 
\label{2dim-N}
\end{aligned}$$ two first order differential equations and become $$\begin{aligned}
        &\varphi'=\frac{\ell}{|\Omega|}\frac{1-\Omega^2\rho^2}{\rho^2}, 
\label{eq-phi}\\
    &{\rho}'^2 = (1-q^2-\Omega^2\rho^2) 
        -\frac{\ell^2(1-\Omega^2\rho^2) }
        {\Omega^2{\rho}^2}.
\label{eq-rho}\end{aligned}$$ We also see that with simply gives $$\begin{aligned}
     z= q\sigma.
\label{zsigma}
\end{aligned}$$ The change of signature of parameters: $$\begin{aligned}
    \ell \to - \ell, \quad q \to - q, \quad\mbox{and}\quad \Omega \to - \Omega,
\end{aligned}$$ can be absorbed by the inversion of coordinates: $$\begin{aligned}
    \varphi \to - \varphi, \quad z \to - z, \quad\mbox{and}\quad t \to - t,
\end{aligned}$$ respectively. Then we restrict ourselves in the case $   \ell \ge 0, q\ge 0$ and $\Omega\ge 0$.

Limiting cases
--------------

Before we obtain general solutions for the stationary rotating strings, we see that two limiting cases of parameters $\ell, q$, (1) $\ell+q = 1$ and (2) $\ell=0$ (see Figs.\[FigHel\] and \[FigPla\]).

### Helical strings 

In the case of $\ell+q 