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A search for the rare decay $\eta^{\prime} \to e^+e^-$ has been performedwith the SND detector at the VEPP-2000 $e^+e^-$ collider. The inverse reaction$e^+e^- \to \eta^{\prime}$ and $\eta^{\prime}$ five decay chains have been usedfor this search. The upper limit $\Gamma_{\eta^{\prime} \to e^+e^-}<0.002$ eVat the 90\% confidence level has been set. A sensitivity of SND in a search for$\eta \to e^+e^-$ decay has been studied. For this perpose we have analyzed adata sample with an integrated luminosity of 108 nb$^{-1}$ collected in thecenter-of-mass energy range 520-580 MeV. There are no background events for thereaction $e^+e^- \to \eta$ with decay $\eta \to \pi^0\pi^0\pi^0$ have beenfound. In the absence of background, a sensitivity to $B(\eta \to e^+e^-)$ of10$^{-6}$ can be reached during two weeks of VEPP-2000 operation.

Search for the decays $\eta^{\prime} \to e^+e^-$ and $\eta \to e^+e^-$ at the VEPP-2000 $e^+e^-$ collider