We define the functions $\tilde{\alpha}_{i}(\mu_{1},\mu_{2},\eta)$ by

(3.4)

$$\begin{cases}\tilde{\alpha}_{1}=\alpha_{1}(\mu_{1},\mu_{2},\eta)+\sum_{n\in\mathbb{Z}\setminus\{0\}}\{\alpha_{1}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{22}}}\}\\ \tilde{\alpha}_{2}=\alpha_{2}(\mu_{1},\mu_{2},\eta)+\sum_{n\in\mathbb{Z}\setminus\{0\}}\{\alpha_{2}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{11}}}\}\\ \tilde{\alpha}_{3}=\alpha_{3}(\mu_{1},\mu_{2},\eta)+\sum_{n\in\mathbb{Z}\setminus\{0\}}\{\alpha_{3}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{11}+2a_{12}+a_{22}}}\}\end{cases}$$

Then $\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{\alpha}_{3}$ are convergent away from $\mathfrak{D}$, 1-periodic in $\eta$, and $\Delta_{a}$-harmonic away from $\mathfrak{D}$. Morever the functions

$$\tilde{v}^{11}=\tilde{\alpha}_{1}+\tilde{\alpha}_{3},\quad\tilde{v}^{22}=\tilde{\alpha}_{2}+\tilde{\alpha}_{3},\quad\tilde{v}^{12}=\tilde{v}^{21}=-\tilde{\alpha}_{3},\quad\tilde{w}=Aa^{ij}\tilde{v}^{ij}$$

provide a solution in the periodic setting to (2.2)(2.3) away from $\mathfrak{D}$, which also solves the distributional equation (2.4) globally.