(First order linearised solution) The equations defining $v^{ij}$ and $w$

(2.9)

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

or equivalently

$$v^{ij}d\mu_{i}\otimes d\mu_{j}=\alpha_{1}d\mu_{1}^{2}+\alpha_{2}d\mu_{2}^{2}+\alpha_{3}(d(\mu_{1}-\mu_{2}))^{2},\quad w=Aa^{ij}v^{ij}$$

provide a solution to the integrability condition (2.2) and the harmonicity condition (2.3) away from $\mathfrak{D}$, which also satisfies the distributional equation (2.4) globally.