TITLE: Psi-Series of Quadratic Vector Fields on the Plane AUTHORS: Amadeu Delshams(1) and Arnau Mir(2) (1) Departament de Matem\`atica Aplicada I, Universitat Polit\`ecnica de Catalunya, Diagonal~647, 08028 Barcelona, Spain. (2) Departament de Matem\`atiques i Inform\`atica, Universitat de les Illes Balears, Crta. de Valldemossa, Km.~7,5, 07071 Palma de Mallorca, Spain. ABSTRACT: Psi-series (i.e., logarithmic series) for the solutions of quadratic vector fields 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 algebraic first integral. Besides, a criterion about non-existence of an analytic first integral is given.