Uniqueness of supersymmetric AdS₅ black holes with SU(2) symmetry

We prove that any supersymmetric solution to five-dimensional minimal gauged supergravity with $SU(2)$ symmetry, is timelike outside an analytic horizon, a Gutowski-Reall black hole or its near-horizon geometry. The proof combines delicate analysis the general form for K\"ahler metric cohomogeneity-1 symmetry. also soliton this theory, symmetry and nut complex bolt, has base enhanced $U(1)\times SU(2)$ we exhibit family of asymptotically AdS$_5/\mathbb{Z}_p$ solitons $p \geq 3$ bolt in class.