I am looking for a function $f:\mathbb C^2 \rightarrow \mathbb C^2$ that satisfies the two equations

$$\partial_{z_2}f_1(z_1,z_2) + \partial_{z_1} f_2(z_1,z_2)=0 \text{ and }$$ $$\partial_{\bar z_1}f_1(z_1,z_2) - \partial_{\bar z_2} f_2(z_1,z_2)=0$$

and in addition, is doubly-periodic in both its complex variables $z_1,z_2$. Does such a function exist and if not, why? I would not even know how to start building such a function.

In particular, I would like to have

$$f_1(z_1+1,z_2)=f_1(z_1,z_2+1)=f_1(z_1,z_2)$$ and

$$f_1(z_1+i,z_2) = e^{2\pi i k_1}f_1(z_1,z_2)$$ and

$$f_1(z_1,z_2+i) = e^{2\pi i k_2}f_1(z_1,z_2)$$

for some fixed $k_1,k_2 \in \mathbb R.$ Please let me know if you do have any questions. I had some typos in there, but hopefully everything is coherent now.