spherical horizon. In fact, Wiseman numerically found that these two families bifurcate at a singular solution with a conic horizon[@Wiseman.T2003a], confirming Kol’s conjecture.

One common feature of these new families of solutions giving rise to non-uniqueness is that they do not have a static and spherically symmetric limit. This feature, together with the uniqueness of asymptotically flat static black holes, strongly indicates that the violation of uniqueness in higher dimensions occurs only when the black horizon is significantly deformed due to high angular momentums or has a non-trivial topology. The main purpose of the present paper is to confirm this observation by searching for regular stationary perturbations of static and maximally symmetric black hole solutions. To be precise, we prove the following theorem.

*For any spherically symmetric vacuum solution that represents a regular black hole spacetime of dimension $d (\ge4)$, a scalar-type perturbation corresponding to a variation of the black hole mass and vector-type perturbations representing rotation of the black hole are the only stationary bounded perturbations that are regular at and outside the horizon (and fall off at infinity in the case $\Lambda\le0$). These exceptional vector perturbations can be parametrised by $[(d-1)/2]$ parameters after identification with background isometries, irrespective 