previous ones, presented in [@Vishneva14]. In our work we take into account the $\pi -a_1$ transition on the external pion line, that has not been done in [@Vishneva14]. On the contrary, we show that the $\pi -a_1$ transition on the virtual axial-vector line, considered in that paper, does not contribute. Thus, we properly account for $\pi -a_1$ mixing effects in the $\tau \to f_1 \pi^{-} \nu_{\tau}$ decay amplitude.

As a result we obtain a reasonable theoretical description of the branching ratio of the $\tau \to f_1\, \pi^{-} \nu_{\tau}$ decay. We believe that as soon as new experimental data on the spectral functions of this process will be available a more detailed test of the extended NJL model will be possible.

The material of this paper is distributed as follows. In Section 2 we establish some convenient notations and review the basic properties of the extended NJL model including its Lagrangian, which is the basis of all our calculations. In Section 3 we give a derivation of the $\tau \to f_1 \pi^{-} \nu_{\tau}$ decay amplitude which does not take into account the $\pi -a_1$ mixing effects. To trace the numerical effect coming out of $\pi-a_1$ transitions, we intentionally delayed this material 