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A sensitivity of the VEPP-2000 $e^+e^-$ collider in a search for the raredecay $\eta \rightarrow e^+ e^-$ has been studied. The inverse reaction $e^+e^- \rightarrow \eta$ is proposed for this search. We have analyzed a datasample with an integrated luminosity of 108 nb$^{-1}$ collected with the SNDdetector in the center-of-mass energy range 520-580 MeV and found no backgroundevents for the reaction $e^+ e^- \rightarrow \eta$ in the decay mode$\eta\to\pi^0\pi^0\pi^0$. In the absence of background, a sensitivity to ${\calB}(\eta \rightarrow e^+ e^-)$ of $10^{-6}$ can be reached during two weeks ofVEPP-2000 operation. Such a sensitivity is better than the current upper limiton ${\cal B}(\eta \rightarrow e^+ e^-)$ by a factor of 2.3.

On a search for the $\eta \rightarrow e^+ e^-$ decay at the VEPP-2000 $e^+e^-$ collider