It is easy to see that for any entire function $f : \mathbb C \to \mathbb C$, its graph $G(f) = \{(z,f(z)) \in \mathbb C^2 \mid z \in \mathbb C\}$ can be translated by $(0,c)$ for any $c \in \mathbb C$, so that all the translated graphs $\{G(f) + (0,c) \mid c \in \mathbb C\}$ form the leaves of a holomorphic foliation of $\mathbb C^2$.
If $V : \mathbb C^2 \to \mathbb C^2$ defines a holomorphic and nowhere zero vector field on $\mathbb C^2$, then there is a unique non-singular holomorphic foliation of $\mathbb C^2$ by holomorphic curves that are everywhere tangent to $V$. These are the orbits of the local $\mathbb C$-action that $V$ induces on $\mathbb C^2$. It is known that the leaves of such a foliation must be topologically either planes or cylinders.
Is it known exactly which holomorphic curves $X \subset \mathbb C^2$ can be a leaf of a non-singular holomorphic foliation of $\mathbb C^2$?
