Black hole and BTZ black string in the Einstein-SU(2) Skyrme model

We present novel analytic hairy black holes with a flat base manifold in the (3+1)-dimensional Einstein SU(2)-Skyrme system with negative cosmological constant. We also construct (3+1)-dimensional black strings in the Einstein $SU(2)$-non linear sigma model theory with negative cosmological constant. The geometry of these black strings is a three-dimensional charged BTZ black hole times a line, without any warp factor. The thermodynamics of these configurations (and its dependence on the discrete hairy parameter) is analyzed in details. A very rich phase diagram emerges.