The process $e^+e^-toetapi^0gamma$ is studied in the center-of-mass energy range 1.05-2.00 GeV using data with an integrated luminosity of 94.5 pb$^{-1}$ collected by the SND detector at the VEPP-2000 $e^+e^-$ collider. The $e^+e^-toetapi^0gamma$ cross section is measured for the first time. It is shown that the dominant mechanism of this reaction is the transition through the $omegaeta$ intermediate state. The measured cross section of the subprocess $e^+e^-toomegaetatoetapi^0gamma$ is consistent with previous measurements in the $e^+e^-topi^+pi^-pi^0eta$ mode. It is found, with a significance of 5.6$sigma$, that the process $e^+e^-toetapi^0gamma$ is not completely described by hadronic vector-pseudoscalar intermediate states. The cross section of this missing contribution, which can originate from radiation processes, e. g. $e^+e^-to a_{0}(1450)gamma$, is measured. It is found to be 15-20 pb in the wide energy range from 1.3 to 1.9 GeV.
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