  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 $\C^2$. The general problem is also reduced to the study of
the combinatorics of certain resolutions of singularities. Whereas we deal with
$\C$-complete vector fields, our results also apply to $\R$-complete ones
thanks to a theorem of Forstneric \cite{forst}.
