The $ u0h_{9/2}$ and $ u0i_{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.
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