up to the Section 4, where we also calculate the spectral distribution of $f_1\pi$ pair and find the two particle decay width of $a_1(1640)\to f_1(1285)\pi$. In Section 5 we summarize our results and make conclusions. In the Appendix we present the integral form of the amplitude describing the $\tau \to f_1 \pi^{-} \nu_{\tau}$ decay and collect some general expressions for the quark-loop-integrals considered.

The quark-meson Lagrangian of the extended NJL model {#sec:1}
====================================================

Let us review the main ingredients of the model which we apply to study the $\tau \to f_1\, \pi^{-} \nu_{\tau}$ decay. Its dynamics is described by the quark Lagrangian density ${\cal L}(x)$ with the effective $U(2)_L\times U(2)_R$ chiral symmetric four-quark interactions $$\begin{aligned}
\label{qint}
{\cal L}(x)&=&\bar q(x) \left(i\hat\partial-m^0 \right)q(x)+{\cal L}_{int}(x), \nonumber \\
{\cal L}_{int}(x)&=&\frac{G_S}{2}\sum_{i=1}^{2}\sum_{a=0}^{3}\left\{[j^S_{a(i)}(x)]^2+[j^P_{a(i)}(x)]^2\right\} \nonumber \\
&-&\frac{G_V}{2}\sum_{i=1}^{2}\sum_{a=0}^{3}\left\{[j^V_{a(i)}(x)]^2+[j^A_{a(i)}(x)]^2\right\},\end{aligned}$$ where $q=(u,d)$ are the up and down current quark fields of mass $m^0=(m^0_u, m^0_d)$; $G_S$ is a coupling determining the strength of the four-quark interactions of scalar and pseudoscalar types, $G_V$ is a coupling of vector and axial-vector interactions. The general form of quark currents is $$j^{S,P,V,A}_{a(i)}(x)=\!\!\int\! d^4x_1 d^4 x_2 \bar q(x_1) F^{S,P,V,A}_{a(i)}(x; x_1,x_2)q(x_2).$$ At each value of index $i$ the sum over $a$ in (\[qint\]) is invariant with respect to 