Planar analytic vector fields with generalized rational first integrals

The main purpose of this paper is to characterize a germ of planar holomorp hic vector field at an elementary singular point having a generalized ratio nal first integral. Our results generalize a result due to Poincare on a ne cessary condition of the existence of a rational first integral for planar polynomial systems. As two applications of our main result, we give the nec essary and sufficient conditions on the existence of rational first integra l for planar quadratic systems having either a weak nondegenerate singular point, or a degenerate elementary singular point. (C) 2001 Editions scienti fiques et medicales Elsevier SAS.