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

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.