We present the first study of the process J/ψ→γηπ0 using (223.7±1.4)×106 J/ψ events accumulated with the BESIII detector at the BEPCII facility. The branching fraction for J/ψ→γηπ0 is measured to be B(J/ψ→γηπ0)=(2.14±0.18(stat)±0.25(syst))×10-5. With a Bayesian approach, the upper limits of the branching fractions B(J/ψ→γa0(980),a0(980)→ηπ0) and B(J/ψ→γa2(1320),a2(1320)→ηπ0) are determined to be 2.5×10-6 and 6.6×10-6 at the 95% confidence level, respectively. All of these measurements are given for the first time.
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