In this work we consider formal singular vector fields in C 3 with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlevé equations (Pj ), j ∈ {I, II, III, IV, V }, for generic values of the parameters. Under generic assumptions we give a complete formal classification for the action of formal diffeomorphisms (by changes of coordinates) fixing the origin and fibered in the independent variable x. We also identify all formal isotropies (self-conjugacies) of the normal forms. In the particular case where the flow preserves a transverse symplectic structure, e.g. for Painlevé equations, we prove that the normalizing map can be chosen to preserve the transverse symplectic form.

In this work which follows directly [Bit16b, Bit16c], we consider analytic singular vector fields in C 3 with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional differential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlevé equations (P j ) j=I...V , for generic values of the parameters. Under suitable assumptions, we provide an analytic classification under the action of fibered diffeomorphisms, based on the study of the Stokes diffeomorphisms obtained by comparing consecutive sectorial normalizing maps à la Martinet-Ramis / Stolovitch [MR82, MR83, Sto96]. These normalizing maps over sectorial domains are obtained in the main theorem of [Bit16c], which is analogous to the classical one due to Hukuhara-Kimura-Matuda [HKM61] for saddle-nodes in C 2 . 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 [Bit16b].

In this work, we consider germs of analytic singular vector fields in C 3 with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional differential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlevé 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 . 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 classification under the action of fibered diffeomorphisms, based on the study of the so-called Stokes diffeomorphisms obtained by comparing consecutive sectorial normalizing maps à la Martinet-Ramis / Stolovitch for 1-resonant vector fields.