  Considering the situation that a single chiral partilce, $\eta$, is initially
emitted, we study the hidden-charm di-eta decays of charmoniumlike state
$Y(4660)$ and the predicted charmonium $\psi(4790)$, i.e., $Y(4660)/\psi(4790)
\to J/\psi\eta\eta$ through the inetermediates,
$\eta\left[D^{(*)}\bar{D}^{(*)}\right]$ and/or $\eta\
\left[D_s^{+(*)}{D}_s^{-(*)}\right]$, and answer to the important question
whether there exist isoscalar charmoniumlike structures in the
$D^{(*)}\bar{D}^{(*)}$ and/or $D_s^{+(*)}{D}_s^{-(*)}$ channels. Our results
predict that there will be enhancement structures near $D\bar{D}^*$,
$D^*\bar{D}^*$ and $D_s\bar{D}_s^*$ thresholds for $Y(4660)$ and near
$D^*\bar{D}^*$, $D_s\bar{D}_s^*$ and $D_s^*\bar{D}_s^*$ thresholds for
$\psi(4790)$ in the $M_{\mathrm{max}}({J/\psi \eta})$ distributions of
$Y(4660)/\psi(4790) \to \eta \eta J/\psi$, respectively. These peaks are
accessible at future experiments, especially BESIII, Belle, BaBar and
forthcoming BelleII.
