In the main result of this paper we give a method to construct all pairs of homogeneous commuting vector fields on ${\mathbb C}^2$ of the same degree $d \geq 2$ (Theorem 1). As an application, we ify, up to linear transformations of ${\mathbb C}^2$, all pairs of commuting homogeneous vector fields on ${\mathbb C}^2$, 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).