Time-reversibility and integrability of p:-q resonant vector fields

TL;DR

This paper investigates the relationship between local integrability and time-reversibility of two-dimensional vector fields near p:-q resonant singular points, extending existing theory from the 1:-1 case to general p:-q cases.

Contribution

It generalizes Sibirsky's theory on time-reversibility and integrability from the 1:-1 resonance to arbitrary p:-q resonances in 2D vector fields.

Findings

Established criteria linking integrability and time-reversibility for p:-q resonant points.

Extended the theoretical framework from 1:-1 to general p:-q resonances.

Provided insights into the structure of resonant vector fields near singularities.

Abstract

We study local analytical integrability in a neighborhood of $p:-q$ resonant singular point of a two-dimensional vector field and its connection to time-reversibility with respect to the non-smooth involution $ \varphi(x,y)=(y^{p/q},x^{q/p}). $ Some generalizations of the theory developed by K.~S.~Sibirsky for $1:-1$ resonant case to the $p:-q$ resonant case are presented.