  We study supersymmetric $AdS_5\times \Sigma$ solutions, for $\Sigma$ being a
topological disk with non-trivial $U(1)$ holonomy at the boundary or
"half-spindle", in seven-dimensional $N=2$ gauged supergravity coupled to three
vector multiplets. We consider compact and non-compact gauge groups,
$SO(4-p,p)$, for $p=0,1,2$, and find a number of $AdS_5\times \Sigma$ solutions
with $SO(2)\times SO(2)$ and $SO(2)_{\text{diag}}$ residual symmetry from
$SO(4)$ and $SO(2,2)$ gauge groups. We also find an $SO(2)_R\subset SO(3)_R\sim
SU(2)_R$ symmetric solution which can be regarded as a solution of pure $N=2$
gauged supergravity with $SO(3)_R$ gauge group and all the fields from vector
multiplets vanishing. The solutions preserve $\frac{1}{2}$ of the original
$N=2$ supersymmetry and could be interpreted as supergravity duals of $N=1$
superconformal field theories in four dimensions. In particular, some of these
solutions can be embedded in ten or eleven dimensions in which a description in
terms of five-branes wrapped on a topological disk can be given.
