  The generalised Hopf equation is the first order nonlinear equation with data
$\Phi$ a holomorphic functions and $\eta\geq 1$ a positive weight, \[
h_w\,\overline{h_\wbar}\,\eta(w) = \Phi.\] The Hopf equation is the special
case $\eta(w)=\tilde{\eta}(h(w))$ and reflects that $h$ is harmonic with
respect to the conformal metric $\sqrt{\tilde{\eta}(z)}|dz|$. This article
obtains conditions on the data to ensure that a solution is open and discrete.
We also prove a strong uniqueness result.
