  By solving the Lippmann-Schwinger equation, possible hadronic molecules in
the $D^*\bar K$-$D\bar K^*$ coupled-channel system are investigated with the
one-meson exchange potentials, where both vector and pseudoscalar mesons are
considered as exchange particles. We find an S-wave virtual state with mass
$2472(18)$ MeV, and a resonance with mass $M=2747(31)$ and width $\Gamma=7(3)$
MeV. In the $D^* \bar K$ invariant mass distribution, the virtual state appears
as a cusp at the $D^*\bar K$ threshold, while the resonance potentially
manifests as a dip. In particular, we take into account the $D\bar K \pi$
three-body dynamics due to the on-shell pion exchange and the finite decay
widths for $D^*$ and $\bar K^*$. Additionally, the cutoff dependence and the
SU(4) breaking effect are investigated in our work. Our results also indicate
that the accurate measurement for the decay width of the $D\bar K^*$ resonance
can help us to evaluate the SU(4) breaking effect in the future.
