inv}}=\frac{1}{(2\pi)^3}\frac{ p_{\bar K^0}
\tilde{p}_{\pi} }{4M_{D^0}^2} \left| t_{ D^0 \to \bar K^0 \pi^+
\pi^-} \right|^2,$$ where $p_{\bar K^0}$ is the $\bar{K}^0$ momentum in the global CM frame ($D^0$ at rest) and $\tilde{p}_{\pi}$ is the pion momentum in the $\pi^+ \pi^-$ rest frame, $$\begin{aligned}
\label{eq:pJpsi}
p_{\bar{K}^0} &=& \frac{\lambda^{1/2}(M_{D^0}^2, M_{\bar K^0}^2,
M_{\rm inv}^2)}{2M_{D^0}}, \\
\tilde{p}_{\pi} &=& \frac{\lambda^{1/2}(M_{\rm inv}^2, m_{\pi}^2,
m_{\pi}^2)}{2M_{\rm inv}}  ,\end{aligned}$$ and similarly for the case of the $\pi^0 \eta$ production.

Before closing this section we should mention that in a three hadron final state one must look for the interaction of three particles, for which one must in principle deal with Faddeev equations [@Faddeev:1960su]. Most of the applications of Faddeev equations are done for three baryon systems but calculations for three mesons are becoming available [@albertokan]. However, for the purpose of the present work it is instructive to follow the idea in Ref. [@dosreis] for the analogous $D^+ \to K^- \pi^+ \pi^+$ reaction. In this work two body unitarity is imposed on the two body systems and diagrams related to three body unitarity are evaluated perturbatively. They are found relevant close to threshold but fade away rapidly of higher energies. What we have done is in this line and we have 