By solving the Lippmann-Schwinger equation, possible hadronic molecules in the $D^{⁎}\overline{K}$- $D{\overline{K}}^{⁎}$ 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 $M=2487$ MeV, and a resonance with $M=2759$ and width $\mathrm{\Gamma }=18$ MeV. In the $D^{⁎}\overline{K}$ invariant mass distribution, the virtual state appears as a cusp at the $D^{⁎}\overline{K}$ threshold, while the resonance potentially manifests as a dip. In particular, we take into account the $D\overline{K}\pi$ three-body dynamics due to the on-shell pion exchange and the finite decay width for ${\overline{K}}^{⁎}$.