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