Doubly-resonant saddle-nodes in $C^3$ and the fixed singularity at infinity in the Painlev{\'e} equations: formal classification

TL;DR

This paper provides a complete formal classification of doubly-resonant saddle-node singularities in three-dimensional complex vector fields, especially those arising from Painlevé equations, including their symmetries and normal forms.

Contribution

It offers a comprehensive formal classification of such singularities, identifying all formal isotropies and normal forms, particularly in the transversely symplectic case relevant to Painlevé equations.

Findings

Complete formal classification of doubly-resonant saddle-nodes.

Identification of all formal isotropies of the normal forms.

Normalizing maps can preserve transversely symplectic structures.

Abstract

In this work we consider formal singular vector fields in $ C^{3}$with an isolated and doubly-resonant singularity of saddle-node typeat the origin. Such vector fields come from irregular two-dimensionalsystems with two opposite non-zero eigenvalues, and appear for instancewhen studying the irregular singularity at infinity in Painlev{\'e} equations$(P\_{j})\_{j\in(I,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 coordinates)fixing 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 Painlev{\'e} equations, we provethat the normalizing map can be chosen to preserve the transversesymplectic form.