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

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 $\phi (x,y)=(y^{p/q},x^{q/p}).$ Some generalizations of the theory developed by K.~S.~Sibirsky for $1:-1$ resonant case to the $p:-q$ resonant case are presented.