Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that for every $f\in\mathbb{C}[X]$ there exists $k\in\mathbb{N}$ such that $V^k(f) = 0$, does it necessarily follow that $V$ is complete? By complete vector field, I mean that the flow $\phi_t(x)$ of $V$ exists for all $x\in X$ and for all $t\in \mathbb{C}$.