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

TL;DR

This paper revisits and completes Bureau's 1981 classification of quadratic differential systems in two variables without movable critical points, correcting and expanding previous results, and explores their geometric and symmetry properties.

Contribution

The authors complete Bureau's classification by adding overlooked cases, correct previous arguments, and analyze the geometric and symmetry aspects of these systems.

Findings

Completed classification of quadratic systems without movable critical points.

Identified and corrected gaps in Bureau's original classification.

Analyzed the birational geometry of associated initial condition spaces.

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.