of the Chern-Simons coupling is larger than some critical value, spherical black holes in such a general EMCS-theory no longer enjoy the uniqueness property. In the present paper, motivated from sting theory, we restrict attention to a special class of EMCS-theory, that is, five-dimensional minimal supergravity and then are able to show the above uniqueness theorem. It would be interesting to find the precise onset of this non-uniqueness property in general EMCS theory, using the formulas developed in this paper.

The rest of the paper is devoted to prove the above uniqueness theorem. In the next section, we present the metric and the gauge potential in Einstein-Maxwell-Chern-Simons theory with three Killing symmetries, introduce the Weyl-Papapetrou coordinates, and reduce the system to a non-linear sigma model with certain symmetries. In Sec. \[sec:mazur\], using the matrix representation of the sigma model, we derive a divergence identity/Mazur identity associated with our nonlinear sigma model. A good part of the material in Sec. \[sec:metric\] and the first part of Sec. \[sec:mazur\] concerning the matrix representation is discussed in [@BCCGSW]. Then, in Sec. \[sec:coincidence\], presenting our boundary conditions for our sigma model fields and using the Mazur identity, we show that if two asymptotically flat 