  The isospin violation process $Y(4260) \to J/\psi \eta \pi^{0}$ is studied
assuming that $Y(4260)$ is a $D_{1} \bar{D}+c.c.$ hadronic molecule. In
association with the production of the $Z_c(3900)$, which is treated as a $D
\bar{D}^{*}+c.c.$ hadronic molecule, this process can help us distinguish their
molecular natures from other scenarios, since the incomplete cancellation
between the charged and neutral--meson loops, which are prominent in the
molecular picture only, produces a peak in the $e^+e^-\to Y(4260)\to
J/\psi\eta\pi^0$ cross section at the $D_{1} \bar{D}+c.c.$ threshold and a very
prominent peak in the $J/\psi \eta$ invariant mass spectrum in between the $D
\bar{D}^{*}+c.c.$ thresholds; the latter being much narrower than the
corresponding one in the isospin conserving channel, i.e. $J/\psi \pi^+
\pi^{-}$. The partial width of $Y(4260)\to J/\psi\eta\pi^0$ is about $4 \times
10^{-4}$ of that of $Y(4260)\to J/\psi\pi^+\pi^-$. The cross section of
$e^+e^-\to Y(4260)\to J/\psi\eta\pi^0$ at the $D_{1} \bar{D}+c.c.$ threshold is
about $0.05 \ \mathrm{pb}$ which is much larger than that produced by the
nearby resonances. These features are the direct consequences of the assumed
nature of these two states which might be accessible at the high-statistics
experiments such as BESIII and LHCb.
