# Doubly-resonant saddle-nodes in \$C^3\$ and the fixed singularity at infinity in the Painlevé equations: formal classification.

* Corresponding author 1 IRMA - Institut de Recherche Mathématique Avancée

Abstract : In this work we consider formal singular vector fields in \$ C^{3}\$with an isolated and doubly-resonant singularity of saddle-node typeat the origin. Such vector fields come from irregular two-dimensionalsystems with two opposite non-zero eigenvalues, and appear for instancewhen studying the irregular singularity at infinity in Painlevé equations\$P {j} {j\inI,II,III,IV,V}\$, for generic values of the parameters.Under generic assumptions we give a complete formal classificationfor the action of formal diffeomorphisms by changes of coordinatesfixing the origin and fibered in the independent variable. Wealso identify all formal isotropies self-conjugacies of the normalforms. In the particular case where the flow preserves a transversesymplectic structure, e.g. for Painlevé equations, we provethat the normalizing map can be chosen to preserve the transversesymplectic form.

Keywords : Irregular singularity Normal form Resonant singularity Singular vector field Painlevé equations

Author: Amaury Bittmann -

Source: https://hal.archives-ouvertes.fr/