contributes to the amplitude as $$\begin{aligned}
&&e^{\mu\nu\lambda}_{\, . \, . \ . \, \alpha} \, p^\alpha_{\pi} \left(2 p_{f_{1}}^{2} + p_\pi p_{f_{1}}\right) - e^{\mu\nu\lambda}_{\, . \, . \ . \, \alpha}\, p^\alpha_{f_{1}} \! \left( p_\pi p_{f_{1}} \right) \nonumber \\
&& + e^{\mu\nu}_{\, . \, .\, \alpha\beta}\, p^\alpha_{\pi} p^\beta_{f_{1}} p_{f_{1}}^{\lambda} - e^{\lambda\mu}_{\, . \, .\, \alpha\beta}\,   p^\alpha_{f_1} p^\beta_{\pi} p_{\pi}^{\nu}.\end{aligned}$$ Due to $\pi -a_1$ transitions this result is multiplied by the 4-momentum $p_{\pi}^{\lambda}$. That vanishes it.

The second term in eq. (\[trace\]) is easily calculated, giving $-8im^2e^{\mu\nu\lambda\alpha} (2k+p_{f_1}-p_\pi )_\alpha$. The result of its integration over $k$ depends on the structure of the integrand. In the case considered, we have three similar structures which differ only by the functions $A_M$ and $B_M$ at the vertices of anomalous triangle diagrams. These alternatives are absorbed by the corresponding coefficient $I_{4}^{A_{f_1}A_{\alpha_1}\dots}$ in the following common structure $$\begin{aligned}
8 m^{4} I_{4}^{A_{f_1}A_{\alpha_1}\dots} e^{\mu\nu\lambda}_{\, . \, . \ . \, \alpha}\, p^\alpha_{f_{1}},\end{aligned}$$ where the integrals $I_{4}^{A_{f_1}A_{\alpha_1}\dots}$ are given by (\[In\]).

Taking into account these remarks, one obtains the corrections induced by the $\pi -a_1$ transitions to the total amplitude of the $\tau \to f_1 \pi^{-} \nu_{\tau}$ decay. As a result the total amplitude is $$\begin{aligned}
\label{Mtot}
&& \mathcal{M}_{tot}  =  G_{F}V_{ud}l^\mu 