Polarized and half polarized U(1) symmetric vacuum spacetimes with AVTD behaviour.

    I sketch, using Kichenassamy - Rendall ideas, a simplified and generalized proof of the Fuchs theorem for differential equations with a singularity. I use the theorem to construct solutions of polarized and half polarized U(1) symmetric vacuum spacetimes with "Asymptotically Velocity Terms Dominated" (AVTD) behaviour near the singularity. I show that the definition I give of half polarization for U(1) symmetric vacuum space-times is a necessary and sufficient condition for non polarized such spacetimes to have this AVTD behaviour.