  We study the $J/\psi \to \gamma \pi^+ \pi^-$, $\gamma \pi^0 \eta$ reactions
from the perspective that they come from the $J/\psi \to \phi(\omega) \pi^+
\pi^-, \rho^0\pi^0 \eta$ reactions, where the $\rho^0$, $\omega$, and $\phi$
get converted into a photon via vector meson dominance. Using models
successfully used previously to study the $J/\psi \to \omega (\phi) \pi\pi$
reactions, we make determinations of the invariant mass distributions for
$\pi^+ \pi^-$ in the regions of the $f_0(500)$, $f_0(980)$, and for $\pi^0
\eta$ in the region of the $a_0(980)$. The integrated differential widths lead
to branching ratios below present upper bounds, but they are sufficiently large
for future check in updated facilities.
