Formes normales semi-classiques des systemes completement integrables au voisinage d'un point critique de l'application moment

The semi-classical study of a 1-dimensional Schr"odinger operator near a non-degenerate maximum of the potential has lead Colin de Verdi`ere and Parisse to prove a microlocal normal form theorem for any 1-dimensional pseudo-differential operator with the same kind of singularity. We present here a generalization of this result to pseudo-differential integrable systems of any finite degree of freedom with a Morse singularity. Our results are based upon Eliasson’s study of critical integrable systems.