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An exact hairy asymptotically locally AdS black hole solution with a flathorizon in the Einstein-nonlinear sigma model system in (3+1) dimensions isconstructed. The ansatz for the nonlinear $SU(2)$ field is regular everywhereand depends explicitly on Killing coordinates, but in such a way that itsenergy-momentum tensor is compatible with a metric with Killing fields. Thesolution is characterized by a discrete parameter which has neither topologicalnor Noether charge associated with it and therefore represents a hair. A $U(1)$gauge field interacting with Einstein gravity can also be included. Thethermodynamics is analyzed. Interestingly, the hairy black hole is alwaysthermodynamically favored with respect to the corresponding black hole withvanishing Pionic field.

Hairy AdS black holes with a toroidal horizon in 4D Einstein-nonlinear $\sigma $-model system