  We study Lorentzian supersymmetric configurations in $D=4$ and $D=5$ gauged
$\mathcal{N}=2$ supergravity. We show that there are smooth $1/2$ BPS solutions
which are asymptotically AdS$_{4}$ and AdS$_{5}$ with a planar boundary, a
compact spacelike direction and with a Wilson line on that circle. There are
solitons where the $S^{1}$ shrinks smoothly to zero in the interior, with a
magnetic flux through the circle determined by the Wilson line, which are AdS
analogues of the Melvin fluxtube. There is also a solution with a constant
gauge field, which is pure AdS. Both solutions preserve half of the
supersymmetries at a special value of the Wilson line. There is a phase
transition between these two saddle-points as a function of the Wilson line
precisely at the supersymmetric point. Thus, the supersymmetric solutions are
degenerate, at least at the supergravity level. We extend this discussion to
one of the Romans solutions in four dimensions when the Euclidean boundary is
$S^{1}\times\Sigma_{g}$ where $\Sigma_{g}$ is a Riemann surface with genus $g >
0$. We speculate that the supersymmetric state of the CFT on the boundary is
dual to a superposition of the two degenerate geometries.
