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The isospin violation process $Y(4260) \to J/\psi \eta \pi^{0}$ is studiedassuming that $Y(4260)$ is a $D_{1} \bar{D}+c.c.$ hadronic molecule. Inassociation 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 theirmolecular natures from other scenarios, since the incomplete cancellationbetween the charged and neutral--meson loops, which are prominent in themolecular picture only, produces a peak in the $e^+e^-\to Y(4260)\toJ/\psi\eta\pi^0$ cross section at the $D_{1} \bar{D}+c.c.$ threshold and a veryprominent peak in the $J/\psi \eta$ invariant mass spectrum in between the $D\bar{D}^{*}+c.c.$ thresholds; the latter being much narrower than thecorresponding 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 $4 \times10^{-4}$ 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 isabout $0.05 \ \mathrm{pb}$ which is much larger than that produced by thenearby resonances. These features are the direct consequences of the assumednature of these two states which might be accessible at the high-statisticsexperiments such as BESIII and LHCb.

Probing the nature of $Y(4260)$ and $Z_c(3900)$ in the isospin violating process $Y(4260) \to J/\psi \eta \pi^0$