ca.classical analysis and odes – Complex-doubly periodic function in two variables?

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

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

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