  We have performed shell-model calculations for the nucleus $^{137}$Xe, which
was recently studied experimentally using the $^{136}$Xe($d,p$) reaction in
inverse kinematics. The main aim of our study has been to investigate the
single-neutron properties of the observed states, focusing attention on the
spectroscopic factors. We have employed a realistic low-momentum two-body
effective interaction derived from the CD-Bonn nucleon-nucleon potential that
has already proved quite successful in describing the spectroscopic properties
of nuclei in the $^{132}$Sn region. Comparison shows that our calculations
reproduce very well the experimental excitation energies and yield
spectroscopic factors that come close to those extracted from the data.
