dependence between $p_1, p_2$ and $p_3$ makes the product to be zero. On the other hand, if pion turns into $a_1$ (the $\pi-a_1$ transition is described by the Lagrangian density $\propto \partial_\nu \vec\pi\vec a_1^\nu$ [@Osipov17je; @Osipov17ap; @Morais17]), the amplitude of the $a_1\to f_1\pi$ transition, which is not zero by itself $e_{\mu\nu\alpha\beta}\epsilon^\mu_{f_1} \epsilon^\nu_{a_1} p_1^\alpha p_2^\beta\neq 0$, will be changed to $\epsilon^\nu_{a_1}\to p^\nu_{\pi}$ and this vanishes the product for the reasons just mentioned above.

Second, we show that it is necessary to take into account the first radially-excited state of the $a_1(1260)$-meson, that is the $a_1(1640)$ resonance. This state has not been considered in [@Vishneva14]. The evidence of this hadron resonance has been recently approved by the new data of COMPASS collaboration [@Wallner17]. To take the $a_1(1640)$ into account, we carry out the calculations in the framework of the extended NJL model [@VolkovW97; @Volkov97; @VolkovE97; @VolkovYu00; @Volkov06; @Volkov16; @Volkov17]. The model allows one to describe both the ground and the first radially-excited meson states in accord with the chiral symmetry requirements.

And finally, we study the influence of the $\pi -a_1$ transitions on the amplitude and demonstrate their significance. Notice the essential difference between present calculations of $\pi -a_1$ effects, and the 