black hole solutions have the same conserved charges, i.e., the mass, electric charge, and two angular momenta, then they must coincide with each other, and complete our proof of the uniqueness theorem. In Sec. \[sec:conclusion\], we summarize our results and discuss possible generalization of our theorem to include non-spherical black objects. We discuss that in order to have a uniqueness theorem for black ring solutions in EMCS-theory, we need to specify rod-data, besides global charges and horizon topology. In Appendix \[sec:A\], we explicitly compute relevant components of the Maxwell-field. In Appendix \[sec:solution\], we provide the five-dimensional Cveti[č]{}-Youm solution with equal charges, and study, in terms of the Weyl-Papapetrou coordinates, the limiting behavior of the solution near relevant boundaries.

Einstein-Maxwell-Chern-Simons system with symmetries {#sec:metric}
====================================================

We consider the bosonic sector of five-dimensional minimal supergravity theory, which can be obtained by a suitable truncation of eleven-dimensional supergravity. The five-dimensional action is given by $$\begin{aligned}
S=\frac{1}{16\pi}
  \left[ 
        \int dx^5\sqrt{-g}\left(R-\frac{1}{4}F^2\right) 
       -\frac{1}{3\sqrt{3}} \int F\wedge F\wedge A 
  \right] \,, 
\label{action} \end{aligned}$$ where we set a Newton constant to be unity and $F=dA$. Varying this action (\[action\]), we derive the Einstein equation $$\begin{aligned}
 R_{\mu \nu } -\frac{1}{2} R g_{\mu \nu } 
 = \frac{1}{2} \left( F_{\mu \lambda } 