Doubly-resonant saddle-nodes in (C^3,0) and the fixed singularity at infinity in Painlev{\'e} equations: analytic classification

Abstract

In this work, we consider germs of analytic singular vector elds in (C^3,0) with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector elds come from irregular two-dimensional dierential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at innity in Painlev{\'e} equations (P j) j=I,...,V for generic values of the parameters. Under suitable assumptions, we prove a theorem of analytic normalization over sectorial domains, analogous to the classical one due to Hukuhara-Kimura-Matuda for saddle-nodes in (C^2,0). We also prove that these maps are in fact the Gevrey-1 sums of the formal normalizing map, the existence of which has been proved in a previous paper. Finally we provide an analytic classication under the action of bered dieomorphisms, based on the study of the so-called Stokes dieomorphisms obtained by comparing consecutive sectorial normalizing maps {\`a} la Martinet-Ramis / Stolovitch for 1-resonant vector fields.