Let $\mathbb{P}^n = \mathbb{C}\mathbb{P}^n$ and let $H_{-d+1}$ be the line bundle with Chern class $-d+1$.
I'm trying to show that a bundle map $\alpha : H_{-d+1} \rightarrow T\mathbb{P}^n$ (i.e. a foliation by curves of degree $d$) is locally described by a polynomial vector field.
To state it more precisely: on affine open sets $U_\beta$, the bundle map $\alpha$ defines a vector field $X_\beta$, (I attempt to describe this correspondence here), and this $X_\beta$ can actually be written as $$X_\beta = \sum_{i=0}^{e+1} X_k + g_{e+1}\sum_{i=1}^n x_i \frac{\partial}{\partial x_i} $$ where each $X_k$ is a homogeneous polynomial vector field of degree $k$ and $g_{e+1}$ is a homogeneous polynomial of degree $e+1$. Where $e$ is some integer $e \leq d-1$.
My attempt: I know, by Chow's theorem, that the integral curves of $X_\beta$ must be algebraic subvarieties of $T\mathbb{P}^n$. I'm wondering if the way to finish the proof is to somehow show that if the integral curves of a vector field are algebraic, then the vector field is polynomial.
Is this true? How can I show it? If this particular strategy doesn't work, could you guide me to a proof of the result stated above?
The result is taken from A foliation of degree different from 1 is determined by its singular scheme by Campillo and Olivares, from the Comptes Rendus.