  The decay $J/\psi \rightarrow \gamma \gamma \phi$ is studied using a sample
of $1.31 \times10^{9}$ $J/\psi$ events collected with the BESIII detector. Two
structures around 1475 MeV/c$^2$ and 1835 MeV/c$^2$ are observed in the $\gamma
\phi$ invariant mass spectrum for the first time. With a fit on the $\gamma
\phi$ invariant mass, which takes into account the interference between the two
structures, and a simple analysis of the angular distribution, the structure
around 1475 MeV/c$^2$ is found to favor an assignment as the $\eta(1475)$ and
the mass and width for the structure around 1835 MeV/c$^2$ are consistent with
the $X(1835)$. The statistical significances of the two structures are $13.5
\sigma$ and $6.3 \sigma$, respectively. The results indicate that both
$\eta(1475)$ and $X(1835)$ contain a sizeable $s\bar{s}$ component.
