Twist of stationary black hole/ring in five dimensions

It is unlikely that uniqueness theorem holds for stationary black holes in higher dimensional spacetimes. However, we will examine the possibility that the higher multipole moments classify vacuum solutions uniquely. Especially, we compute the potentials associated with rotational Killing vectors and look at the dependence on the total mass M and angular momentum J. Consequently, there is a potential $\sigma$ which we cannot write down in terms of integer power of M and J explicitly. This may be regarded as an evidence for the uniqueness using multipole moments generated by $\sigma$.