  The $\nu0h_{9/2}$ and $\nu0i_{13/2}$ strength at $^{137}$Xe, a single neutron
outside the $N=82$ shell closure, has been determined using the
$^{136}$Xe($\alpha$,$^3$He)$^{137}$Xe reaction carried out at 100 MeV. We
confirm the recent observation of the second 13/2$^+$ state and reassess
previous data on the 9/2$^-$ states, obtaining spectroscopic factors. These new
data provide additional constraints on predictions of the same single-neutron
excitations at $^{133}$Sn.
