$p_{\bar K} \to p_{\pi}$, $\tilde{p}_{\pi} \to \tilde{p}_{\eta}$ and $$\label{Eq:19}
\overline{\sum}\sum|t_{f_1,\pi\pi^0\eta}|^2=\frac{1}{3}\vec{k}^{\,2}|\tilde{t}_T|^2\,g_{f_1,K^*\bar K}^2\,g_{K^*,K\pi}^2\,|t_{K\bar K,\pi^0\eta}|^2\, .$$

The value of $g_{K^*,K\pi}$ is taken such that $$\Gamma_{K^*}=\frac{4}{3}\frac{1}{8\pi}\frac{1}{m_{K^*}^2}\tilde{p}_{\pi}^2g_{K^*,K\pi}^2$$ gives the width of the $K^*$, $\Gamma_{K^*}=49.1$ MeV ($g_{K^*,K\pi}=5.5$ MeV).

We should note a small modification in the integral with respect to Ref. [@Aceti:2015zva]. In this latter work a cutoff $\theta(q_{\rm max}-q)$ originated in the chiral unitary approach in the study of the $K\bar K\to \pi \eta$ amplitude was implemented, with $q_{\rm max}=600$ MeV [@Liang:2014tia; @Xie:2014tma]. Since in those works the cutoff is implemented in the center of mass ($\rm CM$) frame, we make a boost of $q$ to the rest frame of the $\pi\eta$ system and implement the cutoff $\theta(q_{\rm max}-q_{\rm CM})$. We show the results obtained in Fig. \[Fig:6\].

![ Differential cross section for $\pi a_0(980)\to\pi\pi^0\eta$ production induced by $f_1(1285)$ excitation. Solid line: considering the triangle diagram of Fig. \[Fig:5\]. Dashed line: using Eq. (\[Eq:5\]) with $\Gamma_R = \Gamma_{f_1(1285)} = 24.1$ MeV, $\Gamma_{R,23}$ a constant and normalizing the curves to the peak of the $f_1(1285)$ (this reflects the shape of the modulus square of the $f_1(1285)$ propagator). The dotted curve is what we would expect for $\pi a_0(980)\to\pi\pi^0\eta$ production from $f_1(1420)$ excitation assuming that the production 