The functions

(2.6)

$$\begin{cases}\alpha_{1}(\mu_{1},\mu_{2},\eta)=&\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\{\frac{1}{2}+\frac{1}{\pi}\arctan(\frac{a_{22}\mu_{2}+a_{12}\mu_{1}}{\sqrt{A}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}})\},\\ \alpha_{2}(\mu_{1},\mu_{2},\eta)=&\frac{1}{2\sqrt{\mu_{2}^{2}+a_{11}|\eta|^{2}}}\{\frac{1}{2}+\frac{1}{\pi}\arctan(\frac{a_{11}\mu_{1}+a_{12}\mu_{2}}{\sqrt{A}\sqrt{\mu_{2}^{2}+a_{11}|\eta|^{2}}})\},\\ \alpha_{3}(\mu_{1},\mu_{2},\eta)=&\frac{1}{2\sqrt{(\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2}}}\\ &\{\frac{1}{2}+\frac{1}{\pi}\arctan(\frac{-a_{11}\mu_{1}-a_{12}\mu_{2}-a_{21}\mu_{1}-a_{22}\mu_{2}}{\sqrt{A}\sqrt{(\mu_{1}-\mu_{2})^{2}+(a_{11}+a_{12}+a_{21}+a_{22})|\eta|^{2}}})\}\end{cases}$$

satisfy the equations on measures

(2.7)

$$\begin{cases}(\Delta_{a}\alpha_{1})d\text{Vol}_{a}=-2\pi\sqrt{A}\int_{\mathfrak{D}_{1}}d\mu_{2},\\ (\Delta_{a}\alpha_{2})d\text{Vol}_{a}=-2\pi\sqrt{A}\int_{\mathfrak{D}_{2}}d\mu_{1},\\ (\Delta_{a}\alpha_{3})d\text{Vol}_{a}=2\pi\sqrt{A}\int_{\mathfrak{D}_{3}}d\mu_{1}\end{cases}$$

where the RHS are signed measures supported on $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$. Morever,

(2.8)

$$\frac{\partial\alpha_{1}}{\partial\mu_{2}}=\frac{\partial\alpha_{2}}{\partial\mu_{1}}=(-\frac{\partial}{\partial\mu_{1}}-\frac{\partial}{\partial\mu_{2}})\alpha_{3}=\frac{\sqrt{A}}{2\pi|\vec{\mu}|_{a}^{2}}.$$

The singularity of $\alpha_{i}$ occurs along $\mathfrak{D}_{i}$ and modulo smooth terms looks like

$$\begin{cases}\alpha_{1}\sim\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}},\\ \alpha_{2}\sim\frac{1}{2\sqrt{\mu_{2}^{2}+a_{11}|\eta|^{2}}},\\ \alpha_{3}\sim\frac{1}{2\sqrt{(\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{12}+a_{22})|\eta|^{2}}}.\end{cases}$$