  There are attempts in the literature to theoretically explain the large
breaking of isotopic invariance in the decay
$\eta(1405)$\,$\to$\,$f_0(980)\pi^0$\,$\to$\,$ 3\pi$ by the mechanism
containing the logarithmic (triangle) singularity, i.e., as being due to the
transition $\eta(1405)\to(K^*\bar K+\bar K^*K)\to(K^+K^-+K^0\bar K^0)\pi^0\to
f_0(980)\pi^0\to3\pi$. The corresponding calculations were fulfilled for a
hypothetic case of the stable $K^*$ meson. Here, we show that the account of
the finite width of the $K^*$ ($\Gamma_{K^*\to K\pi}\approx50$ MeV) smoothes
the logarithmic singularities in the amplitude and results in the suppression
of the calculated decay width $\eta(1405)\to f_0(980) \pi^0\to3\pi$ by the
factor of $6-8$ as compared with the case of $\Gamma_{K^*\to K\pi}$\,=\,0. We
also analyze the difficulties related with the assumption of the dominance of
the $\eta(1405)\to(K^*\bar K+\bar K^*K)\to K\bar K\pi$ decay mechanism and
discuss the possible dynamics of the decay $\eta(1405)\to\eta\pi \pi$. The
decisive improvement of the experimental data on the $K\bar K$, $ K\pi$,
$\eta\pi$, and $\pi\pi$ mass spectra in the decay of the resonance structure
$\eta(1405/1475)$ to $K\bar K\pi$ and $\eta\pi\pi$, and on the shape of the
resonance peaks themselves in the $K\bar K\pi$ and $\eta\pi\pi $ decay channels
is necessary for the further establishing the $\eta(1405)\to3 \pi$ decay
mechanism.
