Real Analytic Vector Fields with First Integral and Separatrices

TL;DR

This paper proves that certain real analytic vector fields with a first integral always have a formal separatrix, but may lack an actual analytic separatrix, highlighting differences between formal and analytic structures.

Contribution

It establishes the existence of a formal separatrix for real analytic vector fields with a first integral and provides a counterexample for the analytic case.

Findings

Existence of a formal separatrix for such vector fields.

Counterexample showing no analytic separatrix necessarily exists.

Clarifies the distinction between formal and analytic separatrices.

Abstract

We prove that a germ of analytic vector field at $(\mathbb{R}^3,0)$ that possesses a non-constant analytic first integral has a real formal separatrix. We provide an example which shows that such a vector field does not necessarily have a real analytic separatrix.