  We used the $^{138}$Ba$(d,\alpha)$ reaction to carry out an in-depth study of
states in $^{136}$Cs, up to around 2.5~MeV. In this work, we place emphasis on
hitherto unobserved states below the first $1^+$ level, which are important in
the context of solar neutrino and fermionic dark matter (FDM) detection in
large-scale xenon experiments. We identify for the first time candidate
metastable states in $^{136}$Cs, which would allow a real-time detection of
solar neutrino and FDM events in xenon detectors, with high background
suppression. Our results are also compared with shell-model calculations
performed with three Hamiltonians that were previously used to evaluate the
nuclear matrix element (NME) for $^{136}$Xe neutrinoless double beta decay. We
find that one of these Hamiltonians, which also systematically underestimates
the NME compared to the others, dramatically fails to describe the observed
low-energy $^{136}$Cs spectrum, while the other two show reasonably good
agreement.
