(1 + G_{\pi^+
\pi^-}t_{\pi^+\pi^- \to \pi^+\pi^-} \nonumber \\
&& + \frac{1}{2} \frac{1}{2} G_{\pi^0\pi^0} t_{\pi^0 \pi^0 \to \pi^+
\pi^-} + \frac{1}{3} \frac{1}{2}G_{\eta \eta} t_{\eta \eta \to \pi^+
\pi^-} \nonumber \\
&& + G_{K^+K^-} t_{K^+ K^- \to \pi^+ \pi^-}), \label{eq:fzero}\end{aligned}$$ and $$\begin{aligned}
 && \!\!\!\!\!\!\!\!\! t(D^0 \to \bar{K}^0 \pi^0 \eta) = V_P
(\sqrt{\frac{2}{3}} + \nonumber \\ && \!\!\!\!\!\!\!\!\!
\sqrt{\frac{2}{3}} G_{\pi^0 \eta} t_{\pi^0 \eta \to \pi^0 \eta}  +
G_{K^+K^-} t_{K^+ K^- \to \pi^0 \eta}), \label{eq:azero}\end{aligned}$$ where $V_P$ is a production vertex, containing the dynamics which is common to all the terms. $G$ is the loop function of two mesons [@npa] and $t_{ij}$ are the transition scattering matrices between pairs of pseudoscalars [@npa]. The $f_0(500)$, $f_0(980)$, and $a_0(980)$ are produced in $s$-wave where $\pi^0
\pi^0$, $\pi^+ \pi^-$ have isospin $I=0$, hence these terms do not contribute to $\pi^0 \eta$ production ($I=1$) in Eq. (\[eq:azero\]). Note that in Eq. (\[eq:fzero\]) we introduce the factor $\frac{1}{2}$ extra for the identity of the particles for $\pi^0 \pi^0$ and $\eta \eta$.

The $t$ matrix is obtained as $$\label{eq:BSeq}
t = [1-VG]^{-1} V,$$ where $V_{ij}$ are the transition potentials evaluated in Refs. [@npa; @danydan]. Explicit expressions for $I=0$ are given in Ref. [@weihong]. We have the $I=1$ case new here 