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There are attempts in the literature to theoretically explain the largebreaking of isotopic invariance in the decay$\eta(1405)$\,$\to$\,$f_0(980)\pi^0$\,$\to$\,$ 3\pi$ by the mechanismcontaining the logarithmic (triangle) singularity, i.e., as being due to thetransition $\eta(1405)\to(K^*\bar K+\bar K^*K)\to(K^+K^-+K^0\bar K^0)\pi^0\tof_0(980)\pi^0\to3\pi$. The corresponding calculations were fulfilled for ahypothetic case of the stable $K^*$ meson. Here, we show that the account ofthe finite width of the $K^*$ ($\Gamma_{K^*\to K\pi}\approx50$ MeV) smoothesthe logarithmic singularities in the amplitude and results in the suppressionof the calculated decay width $\eta(1405)\to f_0(980) \pi^0\to3\pi$ by thefactor of $6-8$ as compared with the case of $\Gamma_{K^*\to K\pi}$\,=\,0. Wealso analyze the difficulties related with the assumption of the dominance ofthe $\eta(1405)\to(K^*\bar K+\bar K^*K)\to K\bar K\pi$ decay mechanism anddiscuss the possible dynamics of the decay $\eta(1405)\to\eta\pi \pi$. Thedecisive 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 theresonance peaks themselves in the $K\bar K\pi$ and $\eta\pi\pi $ decay channelsis necessary for the further establishing the $\eta(1405)\to3 \pi$ decaymechanism.

Isospin breaking decay $\eta(1405) \to f_0(980)\pi^0 \to 3\pi$