Singular SPDEs with the Cauchy-Riemann operator on a torus

We prove the existence of solution to the following ${\mathbb{C}}^3$-valued singular SPDE on the 2D torus ${\mathbb{T}}^2$: \begin{align} \label{CR} \partial_{\bar z} r = r \times \overline{r} + i , γ, {\mathscr W}, \tag{CR} \end{align} where ${\mathrm{\partial }}_{\bar{z}}:=\frac{1}{2}({\mathrm{\partial }}_x+i{\mathrm{\partial }}_y)$ is the Cauchy-Riemann operator on ${\mathbb{T}}^2$, $\mathcal{W}=({\mathcal{W}}_1,{\mathcal{W}}_2,{\mathcal{W}}_3)$ is a real 3D white noise on ${\mathbb{T}}^2$ whose component ${\mathcal{W}}_3$ has zero mean over ${\mathbb{T}}^2$, $\gamma :=({\gamma }_1,{\gamma }_2,{\gamma }_3)$ is an ${\mathbb{R}}^3$-vector and $\gamma \mathcal{W}:=({\gamma }_1{\mathcal{W}}_1,{\gamma }_2{\mathcal{W}}_2,{\gamma }_3{\mathcal{W}}_3)$.