Abstract
In this article, the Galois groupoid of the first Painlevé equation is computed. This computation use E. Cartan's classification of structural equations of pseudogroups acting on
C
2
and the degeneration of the first Painlevé equation on an elliptic equation (
y
′′
=6
y
2
). A definition of reducibility for singular holomorphic foliations is proposed. A characterisation of reducible foliations on their Galois groupoid is given and applied to prove the foliation irreducibility of the first Painlevé equation.