It is easy to see that for any entire function f : ℂ → ℂ, its graph G(f) = {(z,f(z)) ∈ ℂ² | z ∈ ℂ} can be translated by (0,c) for any c ∈ ℂ, so that all the translated graphs {G(f) + (0,c) | c ∈ ℂ} form the leaves of a holomorphic foliation of ℂ².

If V : ℂ² → ℂ² defines a holomorphic and nowhere zero vector field on ℂ², then there is a unique non-singular holomorphic foliation of ℂ² by holomorphic curves that are everywhere tangent to V. These are the orbits of the local ℂ-action that V induces on ℂ². 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 ⊂ ℂ² can be a leaf of a non-singular holomorphic foliation of ℂ²?