Preprint A533/2007
Homogeneous commuting vector fields on $\C^2$.
In the main result of this paper we give a method to construct all pairs of homogeneous commuting vector fields on $\C^2$ of the same degree $d\ge2$ (theorem 1). As an application, we classify, up to linear transformations of $\C^2$, all pairs of commuting homogeneous vector fields on $\C^2$, when $d=2$ and $d=3$ (corollaries 1 and 2). We obtain also necessary conditions in the cases of quasi-homogeneous vector fields and when the degrees are different (theorem 2).