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