For a given analytic function $H$ from $\mathbb{C}$ to $\mathbb{C}$, we define the Polya Vector Field to be $\bar{H}$. This then corresponds to a irrotational, conservative vector field on $\mathbb{C}$.
Therefore in theory if we put a particle in a vector field $\bar{H}$ we should be able to solve for it's trajectory by $\ddot{z(t)}=\bar{H}$. So for $H=-1/z$ (which is like a negative point charge at the origin) I want to solve $\ddot{z(t)}=1/\bar{z}$. I haven't had any luck so far; the conjugate is throwing me off. I tried separating the real and imaginary parts but I am struggling to decouple these two odes. Any help appreciated.
However my main question is, is there anyway to solve $$\ddot{z(t)}=\bar{H}$$ generally?