Regularity and convergence of local first integrals of analytic differential systems

TL;DR

This paper investigates the regularity and convergence of local first integrals in analytic differential systems near singularities, providing new conditions under which these integrals exist and are regular.

Contribution

It establishes the existence of smooth local first integrals in certain resonant cases and characterizes when analytic integrals occur within families of systems.

Findings

Existence of $C^ abla$ smooth first integrals near nonisolated singularities.

In families of systems with the same linear part, integrals are either generically absent or present in a pluripolar subset.

Provides conditions for the regularity and convergence of local first integrals in resonant cases.

Abstract

Poincar\'e proved nonexistence of formal first integrals near a nonresonant singularity of analytic autonomous differential systems. In the resonant case with one zero eigenvalue and others nonresonant, there remains an open problem on regularity and convergence of local first integrals. Here we provide an answer to this problem. The system has always a local $C^\infty$ first integral near the singularity when it is nonisolated. In any finite dimensional space formed by analytic differential systems having the same linear part at the singularity, either all the systems have local analytic first integrals or only the systems in a pluripolar subset have local analytic first integrals.