  We show that there exist supersymmetric solutions of five-dimensional, pure,
$\mathcal{N}=1$ Supergravity such that the norm of the supersymmetric Killing
vector, built out of the Killing spinor, is a real not-everywhere analytic
function such that all its derivatives vanish at a point where the Killing
vector field becomes null. The norm of the Killing vector field then is not an
analytic function on a neighborhood around this point. We explicitly construct
such solutions by using a multi-center Gibbons-Hawking base. Although many of
these solutions have infinite charges, we find explicit examples with finite
charges that asymptote to $AdS_3\times S_2$ and discuss their physical
interpretation.
