is the Ricci one-form. Therefore, there exists the twist potentials $\omega_a$ that satisfy $$\begin{aligned}
 d\omega_a=V_a+\psi_a(3d\mu+\epsilon^{bc}\psi_bd\psi_c)\,.
\label{eq:twistpotential} \end{aligned}$$

Thus, as a consequence of the existence of isometries $\xi_a$, we have eight scalar fields $\lambda_{ab},\omega_a,\psi_a,\mu$ $(a=1,2)$, which we denote collectively by coordinates $\Phi^A=(\lambda_{ab},\omega_a,\psi_a,\mu)$. As we will see soon, other components, such as $a^a{}_i$, $A_i$ are determined by $\Phi^A$. Then, we can find that the equations of motion, eqs. (\[Eineq\]) and (\[Maxeq\]), are cast into a set of equations derived from the following action for sigma-model $\Phi^A$ coupled with three-dimensional gravity with respect to the metric $h_{ij}$, $$\begin{aligned}
S=\int_\Sigma\left({\cal R}^h
 -G_{AB}\frac{\partial \Phi^A}{\partial x^i}
 \frac{\partial \Phi^B}{\partial x^j}h^{ij}\right)\sqrt{|h|}dx^3 \,,\end{aligned}$$ where the target space metric, $G_{AB}$, is given by $$\begin{aligned}
G_{AB}d\Phi^Ad\Phi^B 
&=& \frac{1}{4}{\rm Tr}(\lambda^{-1}d\lambda\lambda^{-1}d\lambda )
   + \frac{1}{4}\tau^{-2}d\tau^2 
   + \frac{3}{2}d\psi^T \lambda^{-1}d\psi \nonumber \\
& &-\frac{1}{2}\tau^{-1}V^T\lambda^{-1}V 
   -\frac{3}{2}\tau^{-1}(d\mu+\epsilon^{ab}\psi_ad\psi_b)^2 \,,\end{aligned}$$ where $\lambda=(\lambda_{ab})$, $\psi=(\psi_1,\psi_2)^T$, $\omega=(\omega_1,\omega_2)^T$ and $V=d\omega-\psi(3d\mu+\epsilon^{bc}\psi_bd\psi_c)$. Varying the action by $h_{ij}$, we obtain the equations $$\begin{aligned}
R^h_{ij} = G_{AB}\frac{\partial \Phi^A}{\partial x^i} 
                 \frac{\partial \Phi^B}{\partial x^j} \,,  
\label{eq:Rij}\end{aligned}$$ where $R^h_{ij}$ denotes the Ricci tensor with respect to $h_{ij}$. Next varying the action by $\Phi^A$, we derive the equation $$\begin{aligned}
\Delta_h\Phi^A+h^{ij}\Gamma^A_{BC}\frac{\partial \Phi^B}{\partial x^i} 
                                  \frac{\partial \Phi^C}{\partial x^j} 
  =0,\end{aligned}$$ where $\Delta_h$ is the Laplacian with respect to the three-dimensional metric $h_{ij}$ and $\Gamma^A_{BC}$ is the 