around 1400 MeV was due to this resonance, but we have given a different interpretation for this feature.

It is interesting to recall here that the triangle singularity of Fig. \[Fig:5\] also shows up in the decay of the $\eta(1405) \to \pi a_0(980)$ which was studied in Refs. [@Wu:2011yx; @Aceti:2012dj], together with the isospin violating $\eta(1405)\to \pi f_0(980)$ decay. This latter decay was abnormally enhanced due to the triangle singularity. In a follow up of Ref. [@Wu:2011yx] in Ref. [@Wu:2012pg] the idea is retaken and applied to study the decay of $J/\psi \to \gamma \eta(1405)(\eta(1475))$ with $\eta(1405) \to K \bar K \pi$, $\eta \pi \pi$ and $3\pi$. In this work the possible contribution of the $f_1(1420)$ in addition to the $\eta(1405)$ was discussed. The $f_1(1420)$ was assumed to be a regular resonance and the triangle singularity enhanced some decay modes, in spite of which its contribution relative to that of the $\eta(1405)$ was found small in the radiative $J/\psi$ decay. Although we find that the $f_1(1420)$ is not a genuine resonance, we have seen that there is indeed strength in the $K \pi \bar K$ and $\pi a_0(980)$ channels in the 1420 MeV region from the decay of the $f_1(1285)$. 