  Using the Nambu--Jona-Lasinio model with the $U(2)\times U(2)$ chiral
symmetric effective four-quark interactions, we derive the amplitude of the
radiative decay $f_1(1285) \to\pi^+\pi^-\gamma$, find the decay width $\Gamma
(f_1\to\pi^+\pi^-\gamma)=346\,\mbox{keV}$ and obtain the spectral di-pion
effective mass distribution. It is shown that in contrast to the majority of
theoretical estimates (which consider the $a_1(1260)$ meson exchange as the
dominant one), the most relevant contribution to this process comes out from
the $\rho^0$-resonance exchange related with the triangle $f_1\rho^0\gamma$
anomaly. The spectral function is obtained to be confronted with the future
empirical data.
