We study holomorphic solutions $f$ of the generalized Dhombres equation
$f(zf(z))=\varphi (f(z))$, $z\in\mathbb C$, where $\varphi$ is
in the class $\mathcal E$ of entire functions. We show, among others, that
there is a nowhere dense set $\mathcal E_{0}\subset\mathcal E$ such that
for every $\varphi\in\mathcal E\setminus\mathcal E_{0}$, any solution
$f$ vanishes at $0$ and hence, satisfies the conditions for local analytic solutions
with fixed point 0 from our recent paper. Consequently, we are able to provide
a characterization of solutions in the typical case where $\varphi\in\mathcal
E\setminus\mathcal E_{0}$. We also show that for polynomial $\varphi$
any holomorphic solution on $\mathbb C\setminus \{ 0\}$ can be extended
to the whole of $\mathbb C$. Using this, in special cases like $\varphi (z)=z^{k+1}$,
$k\in\mathbb N$, we can provide a characterization of the analytic solutions
in $\mathbb C$.