  The two excited vector charmonium states $Y(4220)$ and $Y(4360)$ are
difficult to be understood as pure $c\bar{c}$ charmonium states. Since they are
located close to the mass thresholds of $\bar{D}D_{1}$ and $\bar{D}^*D_{1}$,
they can be viewed as $\bar{D}D_{1}$ and $\bar{D}^*D_{1}$ molecules.
Furthermore, recent studies indicated that the exotic states
$X(3872)$/$Z_c(3900)$ and $X_2(4013)$/$Z_c(4020)$ are the isoscalar/isovector
$\bar{D}D^{*}$ and isoscalar/isovector $\bar{D}^*D^{*}$ molecules,
respectively. In this work, in the molecular picture, we employ the triangle
diagram mechanism to study the productions of $ Z_{c}(3900) $ and $X(3872)$ in
the pionic and radiative decays of $Y(4220)$, as well as their heavy-quark spin
symmetry (HQSS) partners, i.e., the productions of $Z_{c}(4020)$ and
$X_2(4013)$ in the pionic and radiative decays of $Y(4360)$. Using the
effective Lagrangian approach, we obtain the ratios of the branching fractions
$\mathcal{B}[Y(4360) \to Z_c(4020)\pi]/\mathcal{B}[Y(4220)\to
Z_c(3900)\pi]=1.2$ and $\mathcal{B}[Y(4360)\to X_2(4013)
\gamma]/\mathcal{B}[Y(4220)\to X(3872)\gamma]=0.5$, almost independent of model
parameters, which indicate that the productions of $X_2(4013)$ and $Z_c(4020)$
in the radiative and pionic decays of $Y(4360)$ are likely to be measured in
the future. The experimental studies of the predicted decay modes will help
verify the molecular nature of $X(3872)$, $Z_c(3900)$, and $Y(4220)$. We hope
the present work can stimulate experimental and further theoretical studies on
these decay modes.
