  The Belle collaboration recently reported a promising candidate for the
spin-2 $D^*\bar{D}^*$ partner of the $X(3872)$, called the $X_2$ for short,
having a mass of $(4014.3 \pm 4.0 \pm 1.5)~\mathrm{MeV}$ and a width of $(4 \pm
11 \pm 6)~\mathrm{MeV} $. In present work, we assume the $X_2$ as a pure
molecule of the $D^*\bar{D}^*$ under three cases, i.e., pure neutral components
($\theta = 0$), isospin singlet ($\theta = \pi/4$) and neutral components
dominant ($\theta = \pi/6$), where $\theta$ is a phase angle describing the
proportion of neutral and charged constituents. Using an effective Lagrangian
approach, we calculated the partial widths of $X_2\to VV$ and $X_2 \to PP$ ($V$
and $P$ stand for light vector and pseudoscalar mesons, respectively). The
predicted decay widths of $X_2 \to VV$ can reach a few hundreds of
$\mathrm{keV}$, while the decay widths of $X_2 \to PP$ are about several tens
of $\mathrm{keV}$. In addition, the effects from the proportion of neutral and
charged constituent on the decay widths of $X_2\to VV$ and $PP$ are also
investigated. We hope that the present calculations will be checked
experimentally in the future.
