Let $2<q<4$ and $0<\epsilon<q-2$. There is a bounded Green operator for $T^{2}$-invariant functions on the model space with the metric $g_{\text{Taub}}$

$$G_{\text{Taub}}:\{f\in C^{0,\alpha}|f=O(|\vec{\mu}|_{a}^{-q})\text{ for large $|\vec{\mu}|_{a}$}\}\to\{u\in C^{2,\alpha}|u=O(|\vec{\mu}|_{a}^{2-q+\epsilon})\}.$$

such that $u=G_{\text{Taub}}f$ satisfies $\Delta_{\text{Taub}}u=f$.