Analytic integrability for some degenerate planar systems (Q1951455)

The authors prove that an analytic system of ordinary differential equations in a neighborhood of the origin of the form \[ \dot x = y^3 + 3Ax^2y + \cdots,\,\dot y = -x^3 - 3Axy^2 + \cdots \] (where the omitted terms vanish through order three) is analytically integrable if and only if it is formally equivalent to its homogeneous part of degree 3. They also characterize analytic integrability in terms of existence of a formal Lie symmetry and in terms of formal orbital reversibility. They demonstrate that the family contains systems that have a center at the origin yet have no analytic first integral.