Doubly-Resonant Saddle-Nodes in $$\mathbb {C}^{3}$$ C 3 and the Fixed Singularity at Infinity in the Painlevé Equations: Formal Classification

Abstract

In this work we consider formal singular vector fields in $$\mathbb {C}^{3}$$ 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 Painleve equations , $$j\in \left\{ I,II,III,IV,V\right\} $$ , 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 Painleve equations, we prove that the normalizing map can be chosen to preserve the transverse symplectic form.