What is an example of an entire non constant holomorphic function $\gamma: \mathbb{C} \to \mathbb{C}P^2$ such that the image of $\gamma$ is a leaf of a singular holomorphic foliation of $\mathbb{C}P^2$ arising from a non linear polynomial vector field on $\mathbb{R}^2$ or $\mathbb{C}^2$?


Moreover, is it true to say that every leaf of a singular holomorphic foliation of $\mathbb{C}P^2$ is the image of an entire function defined on whole $\mathbb{C}$.


The above question is included in the following post but it did not get an answer. So I ask it as an independent question


[The error in Petrovski and Landis' proof of the 16th Hilbert problem](https://mathoverflow.net/questions/171988/the-error-in-petrovski-and-landis-proof-of-the-16th-hilbert-problem)



The entire parametrization of leaves of singular holomorphic foliation of $\mathbb{C}P^2$
For your first question: $dy=y^2$ can be integrated by quadratures. The solutions are homographies.


For your second question: no it is not true. By the uniformization theorem, the universal covering of a leaf $L$ is either the Riemann sphere $\bar {\mathbb{C}}$, the complex line $\mathbb C$ or the unit disc $\mathbb D$. The latter is the generic situation for generic polynomial foliations. Now, any entire, surjective map $f : \mathbb C\to L$ would realize a holomorphic cover. Because $\mathbb C$ is simply connected, it means that $L$ has $\mathbb C$ for universal cover. 


