I have a coupled PDE in complex variables which I want to solve. But I don't know how to start?
$$\frac{1}{1+\mid z1\mid^2+\mid z2\mid^2}-\frac{\partial y1}{\partial z1}-\frac{z2^*}{z1^*}\frac{\partial y2}{\partial z1}=0,~~\frac{1}{1+\mid z1\mid^2+\mid z2\mid^2}-\frac{\partial y2}{\partial z2}-\frac{z1^*}{z2^*}\frac{\partial y1}{\partial z2}=0$$ Where $*$ denotes complex conjugation and $\mid z\mid^2=zz^*$. So far I only know the solution to the case $z2=0,y2=0$. Which gives
$$\frac{dy1}{dz1}=\frac{1}{1+\mid z1\mid^2}$$ Whose solution is $y1=arctan(z1)+c$. But I have no clue on how to solve the coupled one. Please help?