F_\nu^{ ~ \lambda } 
  - \frac{1}{4} g_{\mu \nu } F_{\rho \sigma } F^{\rho \sigma } \right) \,, 
 \label{Eineq}\end{aligned}$$ and the Maxwell equation $$\begin{aligned}
 d*F+\frac{1}{\sqrt{3}}F\wedge F=0 \,. 
\label{Maxeq}\end{aligned}$$ The purpose of this section is to reduce the above five-dimensional Einstein-Maxwell-Chern-Simons system to a non-linear sigma model with certain target space symmetries. We first consider consequences of the existence of commuting Killing vector fields in our spacetime and identify the target space variables in Subsec. \[subsec:II-A\]. Then, having another (stationary) Killing vector field, we introduce the Weyl-Papapetrou coordinates and write down explicitly the desired non-linear sigma model action in Subsec. \[subsec:II-B\].

Two Killing system {#subsec:II-A}
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Let $\xi_a \ (a=1,2)$ be two mutually commuting Killing vector fields, so that $[\xi_a,\xi_b]=0$, ${\cal L}_{\xi_a} g=0$, and ${\cal L}_{\xi_a}F=0$. Then, introducing the coordinates $x^a$ as Killing parameters of $\xi_a$ (so that $\xi_a = \partial/\partial x^a$), one can express the metric $g$ and the gauge potential one-form $A$, respectively, as $$\begin{aligned}
ds^2 = \lambda_{ab}(dx^a+a^a{}_idx^i)(dx^b+a^b{}_jdx^j) 
      +|\tau|^{-1}h_{ij}dx^idx^j \,, \end{aligned}$$ $$\begin{aligned}
 A = A_adx^a + A_i dx^i \,, \end{aligned}$$ where the functions $\tau:=-{\rm det}(\lambda_{ab})$, $a^a{_i}$, $h_{ij}$, $A_a$, and $A_i$ ($i=3,4,5$) are independent of the coordinates $x^a$.

Let us define the electric one-form $E_a$ with respect to $\xi_a$ by $$\begin{aligned}
