> Has the system of ODEs
$$\frac{dx}{dt}=P(x,y)\\
\frac{dy}{dt}=Q(x,y)
$$
been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of a holomorphic polynomial $F=F(z)$, $z=x+iy$?

I am looking to learn a bit about (complex) ODEs and their interplay with algebraic geometry by some examples, but I couldn't find anything on this special case in Ilyashenko's survey on Hilbert 16 (I guess this case is too special and/or not very interesting as far as Hilbert 16 is concerned).

Nontheless, it seems very natural. If we set $\gamma(t)=x(t)+iy(t)$, this amounts to the equation
$$
\int_{\gamma_t}\frac{dz}{F(z)}=t
$$
where $\gamma_t$ is the curve $\gamma$ "truncated" at $t$ and the RHS is in particular **real**. This can be taken further, for example by assuming $\gamma$ is closed and using the residue theorem to obtain constraints on (the coefficients of) $F$.