Static black holes without spatial isometries: from AdS multipoles to electrovacuum scalarization

We review recent results on the existence of static black holes without spatial isometries in four spacetime dimensions and propose a general framework for their study. These configurations are regular on and outside a horizon of spherical topology. Two different mechanisms allowing for their existence are identified. The first one relies on the presence of a solitonic limit of the black holes; when the solitons have no spatial isometries, the black holes, being a non-linear bound state between the solitons and a horizon, inherit this property. The second one is related to black hole scalarization, and the existence of zero modes of the scalar field without isometries around a spherical horizon. When the zero modes have no spatial isometries, the backreaction of their non-linear continuation makes the scalarized black holes inherit the absence of spatial continuous symmetries. A number of general features of the solutions are discussed together with possible generalizations.