A sensitivity of the VEPP-2000 $e^+e^-$ collider in a search for the rare decay $eta rightarrow e^+ e^-$ has been studied. The inverse reaction $e^+ e^- rightarrow eta$ is proposed for this search. We have analyzed a data sample with an integrated lu
minosity of 108 nb$^{-1}$ collected with the SND detector in the center-of-mass energy range 520-580 MeV and found no background events for the reaction $e^+ e^- rightarrow eta$ in the decay mode $etatopi^0pi^0pi^0$. In the absence of background, a sensitivity to ${cal B}(eta rightarrow e^+ e^-)$ of $10^{-6}$ can be reached during two weeks of VEPP-2000 operation. Such a sensitivity is better than the current upper limit on ${cal B}(eta rightarrow e^+ e^-)$ by a factor of 2.3.
A search for the process $e^+e^- to eta^prime$ has been performed with the SND detector at the VEPP-2000 $e^+e^-$ collider. The data were accumulated at the center-of-mass energy of $957.78pm 0.06$ MeV with an integrated luminosity of about 2.9 pb$^{
-1}$. For reconstruction of the $eta^prime$ meson five decay chains have been used: $eta^{prime}toetapi^+pi^-$ followed by the $eta$ decays to $gammagamma$ and $3pi^0 $, and $eta^{prime} to etapi^0pi^0$ followed by the $eta$ decays to $pi^+pi^-pi^0$, $gammagamma$, and $3pi^0$. As a result, the upper limit has been set on the $eta^prime$ electronic width: $Gamma_{eta^{prime}to e^+e^-} < 0.0020$ eV at the 90% confidence level.
In the experiment with the SND detector at the VEPP-2000 $e^+e^-$ collider the cross section for the process $e^+e^-toetapi^+pi^-$ has been measured in the center-of-mass energy range from 1.22 to 2.00 GeV. Obtained results are in agreement with prev
ious measurements and have better accuracy. The energy dependence of the $e^+e^-toetapi^+pi^-$ cross section has been fitted with the vector-meson dominance model. From this fit the product of the branching fractions $B(rho(1450)toetapi^+pi^-)B(rho(1450)to e^+e^-)$ has been extracted and compared with the same products for $rho(1450)toomegapi^0$ and $rho(1450)topi^+pi^-$ decays. The obtained cross section data have been also used to test the conservation of vector current hypothesis.
The $e^+e^-toetagamma$ cross section has been measured in the center-of-mass energy range 1.07--2.00 GeV using the decay mode $etato 3pi^0$, $pi^0to gammagamma$. The analysis is based on 36 pb$^{-1}$ of integrated luminosity collected with the SND de
tector at the VEPP-2000 $e^+e^-$ collider. The measured cross section of about 35 pb at 1.5 GeV is explained by decays of the $rho(1450)$ and $phi(1680)$ resonances.
