(p,q=4,5)$, where $\rho^2=-{\rm det}(g_{IJ})$. Note that the function $\rho$ is globally well-defined [@Chru08]. That $\rho$ is a harmonic function can be seen by looking at the $(tt)$-component of eq. (\[eq:Rij\]), which is written $$\begin{aligned}
 R_{tt}=\rho \hat D^2\rho=0 \,,\end{aligned}$$ where $\hat D_p$ is the covariant derivative associated with the two-dimensional metric $h_{pq}$. Let $z$ be harmonic function conjugate to $\rho$ which satisfies $\hat D^2 z=0$, $\hat D_p \rho\hat D^pz=0$, $\hat D_p \rho\hat D^p\rho=\hat D_p z\hat D^p z$. Choose the coordinates $(x^4,x^5)$ as $x^4=\rho$ and $x^5=z$. Then, the metric can be written in the Weyl-Papapetrou type form as $$\begin{aligned}
ds^2&=&\lambda_{\phi\phi}(d\phi+a^\phi{}_tdt)^2+\lambda_{\psi\psi}(d\psi+a^\psi{}_tdt)^2 \nonumber\\
&&+2\lambda_{\phi\psi}(d\phi+a^\phi{}_tdt)(d\psi+a^\psi{}_tdt)+|\tau|^{-1}[e^{2\sigma}(d\rho^2+dz^2)-\rho^2 dt^2] \,, \end{aligned}$$ where all the metric components depend only on $\rho$ and $z$.

In this coordinate system, $\Phi^A$ are determined by the equations of motion $$\begin{aligned}
\Delta_\gamma \Phi^A + 
\Gamma^A_{BC}[\Phi^B_{,\rho}\Phi^C_{,\rho}+\Phi^C_{,z}\Phi^C_{,z}]=0 \,, 
\label{eq:scalar}\end{aligned}$$ where $\Delta_\gamma$ is the Laplacian with respect to the abstract three-dimensional metric $\gamma=d\rho^2+dz^2+\rho^2d\varphi^2$. On the other hand, once $\Phi^A$ are given, one can completely determine $\sigma$, $a^\phi{}_t$, $a^\psi{}_t$, $A_i$. In fact, the function $\sigma$ is determined by $$\begin{aligned}
\frac{2}{\rho}\sigma_{,\rho}&=&R^h_{\rho\rho}-R^h_{zz}\nonumber\\
 &=&G_{AB}[\Phi^A_{,\rho}\Phi^B_{,\rho}-\Phi^A_{,z}\Phi^B_{,z}] \,,\\
   \frac{1}{\rho}\sigma_{,z}&=&R^h_{\rho z}\nonumber\\
                            &=&G_{AB}\Phi^A_{,\rho}\Phi^B_{,z} \,.\end{aligned}$$ The integrability $\sigma_{,\rho z}=\sigma_{,z \rho }$ is assured by eq. (\[eq:scalar\]). From eq.(\[eq:twistpotential\]), the metric functions $a^a{_t}$ are determined by $$\begin{aligned}
a^a{}_{t,\rho}
