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We consider $3$-dimensional isolated horizons (IHs) generated by null curvesthat form nontrivial $U(1)$ bundles. We find a natural interplay between the IHgeometry and the $U(1)$-bundle geometry. In this context we consider the Petrovtype D equation introduced and studied in previous works\cite{DLP1,DLP2,LS,DKLS1}. From the $4$-dimensional spacetime point of view,solutions to that equation define isolated horizons embeddable in vacuumspacetimes (with cosmological constant) as Killing horizons to the second ordersuch that the spacetime Weyl tensor at the horizon is of the Petrov type D.From the point of view of the $U(1)$-bundle structure, the equation couples a$U(1)$-connection, a metric tensor defined on the base manifold and the surfacegravity in a very nontrivial way. We focus on the $U(1)$-bundles over$2$-dimensional manifolds diffeomorphic to $2$-sphere. We have derived all theaxisymmetric solutions to the Petrov type D equation. For a fixed value of thecosmological constant they set a $3$-dimensional family as one could expect. Asurprising result is, that generically our horizons are not embeddable in theknown exact solutions to Einstein's equations. It means that among the exacttype D spacetimes there exists a new family of spacetimes that generalize theproperties of the Kerr- (anti) de Sitter black holes on one hand and theTaub-NUT spacetimes on the other hand.

Petrov type D equation on horizons of nontrivial bundle topology