  We present the first study of the process $J/\psi \rightarrow
\gamma\eta\pi^{0}$ using $(223.7\pm1.4)\times10^{6}$ $J/\psi$ events
accumulated with the BESIII detector at the BEPCII facility. The branching
fraction for $J/\psi \rightarrow \gamma\eta\pi^{0}$ is measured to be
$\mathcal{B}(J/\psi \rightarrow \gamma\eta\pi^{0})
=(2.14\pm0.18(stat)\pm0.25(syst))\times10^{-5}$. With a Bayesian approach, the
upper limits of the branching fractions $\mathcal{B}(J/\psi \rightarrow \gamma
a_{0}(980),a_0(980)\rightarrow\eta\pi^0)$ and $\mathcal{B}(J/\psi \rightarrow
\gamma a_{2}(1320),a_2(1320)\rightarrow\eta\pi^0)$ are determined to be
$2.5\times10^{-6}$ and $6.6\times10^{-6}$ at the 95\% confidence level,
respectively. All of these measurements are given for the first time.
