Search for the $X(2370)$ and observation of $eta_{c}toetaetaeta^prime$ in $J/psitogammaetaetaeta^{prime}$

Using a sample of $1.31times10^{9} ~J/psi$ events collected with the BESIII detector, we perform a study of $J/psitogammaetaetaeta^{prime}$ to search for the $X(2370)$ and $eta_{c}$ in the $etaetaeta^{prime}$ invariant mass distribution. No significant signal for the $X(2370)$ is observed, and we set an upper limit for the product branching fraction of ${cal B}(J/psitogamma X(2370)cdot{cal B}(X(2370)toetaetaeta^{prime}) < 9.2times10^{-6}$ at the 90% confidence level. A clear $eta_{c}$ signal is observed for the first time, yielding a product branching fraction of ${cal B}(J/psito gammaeta_{c})cdot{cal B}(eta_{c}to etaetaeta^{prime}) = (4.86pm0.62~({rm stat.})pm0.45~({rm sys.}))times10^{-5}$.