On Bureau's classification of quadratic differential equations in two variables free of movable critical points

Abstract

As part of the efforts aimed at extending Painlev\'e and Gambier's work on second-order equations in one variable to first-order ones in two, in 1981, Bureau classified the systems of ordinary quadratic differential equations in two variables which are free of movable critical points (which have the Painlev\'e Property). We revisit this classification, which we complete by adding some cases overlooked by Bureau, and by correcting some of his arguments. We also simplify the canonical forms of some systems, bring the natural symmetries of others into their study, and investigate the birational equivalence among some of the systems in the class. Lastly, we study the birational geometry of Okamoto's space of initial conditions for Bureau's system VIII, in order to establish the sufficiency of some necessary conditions for the absence of movable critical points.