the stationarity of the strings breaks down at the light cylinder $\rho=\rho_{max}=1/\Omega$. Since the point $\rho=1/\Omega$ is curvature singularity, the geodesics have end points there. This case is a special case of Burden’s solution[@Burden] and discussed in detail by Frolov et al[@FSZH]. The shape of strings in $x-y$ plane is shown in Fig.5.

![Strings in $x-y$ plane: $q=0$. The dashed circle in the left figure is the light cylinder $\rho = 1/\Omega$.\
[]{data-label="FigHor"}](equatorial.eps){width="8cm"}

Both of the functions $\rho(\sigma)$ and $\varphi(\sigma)$ in the solution and are periodic with respect to $\sigma$, but the periods are different. It is clear that $$\begin{aligned}
    \rho(\sigma + \sigma_p ) = \rho (\sigma), \end{aligned}$$ where $\sigma_p = \pi/\Omega$. In this period of $\sigma$, $\varphi$ varies as $$\begin{aligned}
    {\varphi}(\sigma +\sigma_p  ) 
        = {\varphi}(\sigma) + \pi(1-\ell). 
\label{Leg1}\end{aligned}$$ Suppose that $\ell$ is a rational number which can be expressed as $\ell = a/b$, where $a$ and $b$ are relatively prime. During $\sigma$ increases by $n$ times $\sigma_p$, $\varphi$ varies as $$\begin{aligned}
{\varphi}(\sigma +n\sigma_p  ) 
        &= {\varphi} (\sigma) + n\pi(1-\ell) \cr
        &= {\varphi} (\sigma) + n\frac{b-a}{b}\pi. 
\label{Leg2}\end{aligned}$$ If the second term in the right hand side holds $$\begin{aligned}
    n\frac{b-a}{b}\pi= 2\pi m, \quad(n,m : \mbox{natural numbers})\end{aligned}$$ $\rho$ and $\varphi$ have a 