exchange alone gives a larger number Br$\ =8.43\times 10^{-4}$. The sum of these two contributions is Br$\ =3.34\times 10^{-4}$. The radially excited state increase this value up to Br$\ =6.04\times 10^{-4}$. And finally taking into account the $\pi  -a_1$ transitions we come to the final number Br$\ =3.98\times 10^{-4}$ which agrees with presently known empirical values.

Our result shows that both $a_1(1260)$ and $a_1(1640)$ exchanges are very important for the description of data, and it indicates that many ideas about description of the first radial exited meson states in the NJL model seems to be correct. Let us stress that we did not introduce any new parameters to get the consistent result. All model parameters were fixed with the use of other input data, which are not related with the process $\tau\to f_1(1285)\,\pi^- \nu_\tau$. Note, that this is not the only case where the model predictions correspond to the empirical values. We refer to the recent review [@Volkov17], which contains many other examples. Those include the meson production processes in $e^+e^-$ collisions and tau decay modes.

Of some relevance may be the fact that our result does not leave the place for the contribution of a new axial-vector resonance $a_1(1420)$ 