Two types of hairy black holes in the (3+1)-dimensional $O(3)$ sigma-model

We construct stationary axially symmetric black holes with scalar hair in the gravitating nonlinear $O(3)$ sigma model with a partly negative symmetry-breaking potential. Our results show that it supports two qualitatively different types of asymptotically flat hairy black holes, distinguished by their connection to the corresponding solitonic branches and by the structure of their scalar hair; the type I black holes are connected to the vacuum branch and type II black holes are connected to regular scalaron-supported spinning solitons. The type II black holes have a strongly deformed scalar field structure, with the scalar field taking its vacuum value on the symmetry axis and approaching the anti-vacuum in the near-horizon equatorial region. We analyze their global charges, horizon properties, and dependence on the model parameters, comparing the solutions with Kerr black holes. For sufficiently weak gravitational coupling, the branch connectivity changes: the fundamental branches of types I and II reconnect to form one continuous family, while their corresponding second branches form another. We find that, in contrast to the vacuum-connected type I family, the scalaron-supported type II black holes extend to substantially larger horizon radii while preserving a nontrivial scalar configuration outside the horizon.