mixing angles follow from the meson mass formulae. They are $$\begin{aligned}
\label{angles}
&& \theta_{f_{1}} = \theta_{a_{1}} = 81.8^{\circ}, \quad \theta_{\pi} = 59.48^{\circ},  \nonumber\\
&& \theta_{f_{1}}^{0} = \theta_{a_{1}}^{0} = 61.5^{\circ}, \quad \theta_{\pi}^{0} = 59.12^{\circ}.  \end{aligned}$$ Notice, that all parameters of the model are determined from the empirical data, which are not related with the characteristics of the processes considered in this work. Therefore, our estimations can be used to test the predictive power of the extended NJL model.

The decay amplitude without $\pi -a_{1}$ transitions {#sec:2}
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The decay amplitude of the process $\tau \to f_1(1285)\, \pi^- \nu_\tau$ is described by two types of diagrams which are shown in Fig.\[Contact\], and Fig.\[Intermediate\] (the $\pi -a_1$ transitions are neglected there). The first diagram describes the so-called direct contribution to the amplitude. This means that $W$-boson decays directly to the final products of the reaction, $f_1\pi$ - pair, i.e. without a resonance exchange. The latter is taken into account by the second diagram. Notice, that hadrons are alway interact through the one-quark-loop in accord with our action (\[Lagrangiane\]). In addition, we would like to point out that these loop integrals are expanded in momenta of external fields, and only the divergent parts are 