Study of the process $e^+e^- \to \eta\eta\gamma$ in the energy range $\sqrt{s} = \mbox{1.17--2.00}$ GeV with the SND detector

The process $e^+e^-\to\eta\eta\gamma$ is studied in the center-of-mass energy range 1.17-2.00 GeV using data with an integrated luminosity of 201 pb$^{-1}$collected by the SND detector at the VEPP-2000 $e^+e^-$ collider. The $e^+e^-\to\eta\eta\gamma$ cross section is measured for the first time. It is shown that the dominant mechanism of this reaction is the transition through the $\phi\eta$ intermediate state. Our result on the $e^+e^-\to\eta\eta\gamma$ cross section is consistent with the $e^+e^-\to\phi\eta$ measurement in the $\phi\to K^+ K^-$ mode. The search for radiative processes contributing to the $e^+e^-\to\eta\eta\gamma$ cross section is performed, and no significant signal is observed.