  Using a sample of $(225.2\pm 2.8)\times 10^6$ $J/\psi$ events collected with
the BESIII detector, we present results of a study of $J/\psi\rightarrow
\gamma\Lambda\bar{\Lambda}$ and report the first observation of the isospin
violating decay $J/\psi\rightarrow\Lambda\bar{\Sigma}^{0}+c.c.$, in which
$\bar{\Sigma}^{0}$ decays to $\gamma \bar{\Lambda}$. The measured branching
fractions are $\mathcal{B}(J/\psi\rightarrow\bar{\Lambda}\Sigma^{0}$) =
$(1.46\pm0.11\pm0.12) \times10^{-5}$ and
$\mathcal{B}(J/\psi\rightarrow\Lambda\bar{\Sigma^{0}}$) = $(1.37\pm0.12\pm0.11)
\times10^{-5}$. We search for $\Lambda(1520) \rightarrow \gamma \Lambda$ decay,
and find no evident signal, and an upper limit for the product branching
fraction $\mathcal{B}(J/\psi\rightarrow\Lambda\bar{\Lambda}(1520)+c.c.)\times
\mathcal{B}(\Lambda(1520)\rightarrow\gamma\Lambda)<4.1 \times10^{-6}$ is set at
the 90% confidence level. We also report the observation of
$\eta_{c}\rightarrow\Lambda\bar{\Lambda}$ in $J/\psi \rightarrow \gamma
\eta_{c}$, $\eta_{c}\rightarrow\Lambda\bar{\Lambda}$ and measure the branching
fraction $\mathcal{B}(\eta_{c}\rightarrow\Lambda\bar{\Lambda}) =(1.16\pm0.12
(stat)\pm0.19(syst)\pm0.28(PDG))\times10^{-3}$.
