  In a recent theoretical work, Phys. Lett. B785, 112 (2018), we proposed that
a $K^*$ resonance with hidden charm content arises from the $KD\bar D^*$
dynamics, where the $D\bar D^*$ system is treated as a $Z_c(3900)$ or
$X(3872)$. With the motivation of determining its further properties, which can
be observed in experiments, we now present a calculation of the decay processes
of this $K^*$, namely $K^*(4307)$, to two-body channels. Particularly, we
consider the decay channels $J/\psi K^*(892)$, $\bar{D}D_s$, $\bar{D}D_s^*$ and
$\bar{D}^* D_s^*$. The mechanisms of the decay to these channels involve
triangular loops and are a consequence of the internal structure of the state.
Thus, the values found for the decay widths of the proposed $K^*(4307)$ are
related to its nature and should be valuable for carrying on an experimental
study of the $K^*(4307)$. A $K^*$ state with such a mass (in the charmonium
region) and quantum numbers is a clear manifestation of an exotic meson, since,
having hidden charm (i.e., a $c\bar c$ pair), its mass and quantum numbers can
not be explained within a quark-antiquark description.
