Local First Integrals for Analytic Anti-periodic Vector Fields

Abstract Poincaré [Rend. Circ. Mat. Palermo 5 (1891) 161-191; 11 (1897) 193-239] first proposed a criterion about the non-existence of local formal first integrals near a singularity via resonance relations. In this paper, I introduce the concept of first integrals for anti-periodic vector fields and propose some Poincaré type results for identifying the absence or partial existence of local formal first integrals of anti-periodic vector fields. Compared with the previous results, a new algebraic structure for resonance conditions, which is a semimodule over the semiring of natural numbers, is found.