appears worth while to estimate the decay width of the state, that dominates the spectral function. The amplitude of the $a_1(1640)\to f_1(1285)\pi$ decay is given by $$\begin{aligned}
\mathcal{M} & = & 4img_\pi e_{\mu\nu\alpha\beta }\, \epsilon^{\mu}(p_{a_{1}})\epsilon^{\nu}(p_{f_{1}})  p^\alpha_{f_{1}}p^\beta_{\pi}  \nonumber \\
&\times& \left\{I_{3}^{A_{f_{1}}B_{a_{1}}} - \frac{C_{a_1}6m^4}{g_{a_{1} M_{a_1}^2}} I_{4}^{A_{f_{1}}B_{a_{1}}A_{a_{1}}} \right\}. \end{aligned}$$ It follows then that $$\Gamma \left[a_1(1640)\to f_1(1285)\pi \right]=14.1\,\mbox{MeV}.$$ The future measurements will show how good is the extended NJL model in its predictions here. Probably, the tau decay mode studied in this work can serve us with a detailed information on the nature of $a_1(1640)$ state.

Conclusions
===========

The main purpose of our calculations was to apply the extended NJL model to the decay $\tau\to f_1(1285)\,\pi^- \nu_\tau$. Presently available phenomenological data on this mode give a rare opportunity to test the model. As a result we have found that hadronic part of the amplitude is sensitive to the four types of different contributions which are equally important. These are the contact interaction, the exchange by the $a_1(1260)$ meson, the exchange by the first radially excited state, $a_1(1640)$, and the pion creation by the intermediate $a_1(1260)$ meson (the $\pi  a_1$-transitions). Indeed, the contact term alone gives the value Br$\ =2.25\times 10^{-4}$. The $a_1(1260)$ ground state 