Psi-Series of Quadratic Vector Fields on the Plane

Psi series i e logarithmic series for the solutions of quadratic vec tor elds on the plane are considered Its existence and convergence is studied and an algorithm for the location of logarithmic singularities is developed Moreover the relationship between psi series and non integrability is stressed and in particular it is proved that quadratic systems with psi series that are not Laurent series do not have an al gebraic rst integral Besides a criterion about non existence of an analytic rst integral is given Introduction The study of the representation of the solutions of a system of di erential equations and the relationship with the integrability of this system has been considered by many people This relationship was initiated by Painlev e who gave a test of integrability based on the search of systems such that all its solutions could be represented as Laurent series of the time parameter Inc Hil This Painlev e test was later on transformed in an algorithm the ARS algorithm ARS that has been used succesfully to detect integrable systems see RGB for a general review According to this Painlev e principle those systems such that all their so lutions can be represented in terms of Laurent series are in fact integrable