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A search for the rare decay $\eta \rightarrow e^+e^-$ is performed using theinverse process $e^+e^- \rightarrow \eta$ in the decay mode$\eta\to\pi^0\pi^0\pi^0 $. We analyze data with an integrated luminosity of 654nb$^{-1} $ accumulated at the VEPP-2000 $e^+e^-$ collider with the SND detectorat the center-of-mass energy $E=m_\eta c ^2\approx 548 $ MeV, and set the upperlimit $ {\cal B} (\eta \rightarrow e^+ e^-) < 7 \times 10^{-7} $ at the 90\%confidence level.

Search for the process $\mathbf{e^+e^- \to\eta}$