  In this work we make a systematic analysis of the correlated processes
$J/\psi\to \gamma \eta(1440)/f_1(1420)$ with $\eta(1440)/f_1(1420)\to
K\bar{K}\pi$, $\eta\pi\pi$ and $3\pi$, where the role played by the so-called
"triangle singularity mechanism" (TSM) is clarified. Our results agree well
with the experimental data and suggest a small fraction of $f_1(1420)$
contributions in these processes. This study confirms our conclusion in [Phys.
Rev. Lett. 108, 081803 (2012)] that the dynamic feature of the TSM can be
recognized by the strong narrow peak observed in the $\pi\pi$ invariant mass
spectrum of $\eta(1440)\to 3\pi$ with anomalously large isospin violations.
Nevertheless, we explicitly demonstrate that the TSM can produce obvious peak
position shifts for the same $\eta(1440)$ or $f_1(1420)$ state in different
decay channels. This is a strong evidence that the $\eta(1405)$ and
$\eta(1475)$ are actually the same state, i.e. $\eta(1440)$. We also make an
analysis of the radiative decays of $\eta(1440)\to \gamma V$ ($V=\phi$,
$\rho^0$ or $\omega$) which shows that such a one-state prescription seems not
to have a conflict with the so-far existing experimental data. Our analysis may
shed a light on the long-standing puzzling question on the nature of
$\eta(1405)$ and $\eta(1475)$.
