Birationally integrable vector fields on complex projective surfaces

A rational vector field on a complex projective smooth surface $S$ is said to be birationally integrable if it generates, by integration, a one-parameter subgroup of the group $\operatorname{Bir}(S)$ of birational transformations of $S$. We prove that every birationally integrable vector field is regularizable, i.e. birationally conjugated to a holomorphic vector field. Next, we extend this result to any finite-dimensional Lie algebra $\mathfrak g$ of birationally integrable vector fields. This implies that $\mathfrak g$ is naturally included into the Lie algebra of an algebraic subgroup of $\operatorname{Bir}(S)$. Moreover, we obtain a complete birational classification of birationally integrable Lie algebras that are of dimension two or semisimple, exhibiting holomorphic normal forms of them. We also characterize those birationally integrable algebras of rational vector fields that are maximal.