Complete polynomial vector fields on ${\text{C}}^2$, Part I

In this work, under a mild assumption, we give the classification of the complete polynomial vector fields in two variables up to algebraic automorphisms of ${\text{C}}^2$. The general problem is also reduced to the study of the combinatorics of certain resolutions of singularities. Whereas we deal with $\text{C}$-complete vector fields, our results also apply to $\mathbb{R}$-complete ones thanks to a theorem of Forstneric \cite{forst}.