Rational curves and ordinary differential equations

TL;DR

This paper characterizes complex second order ODE systems whose solutions form rational curves, identifying a differential invariant that signals integrability and reveals an infinite family of such systems.

Contribution

It introduces an explicit differential invariant that characterizes these ODE systems and demonstrates their integrability and infinite diversity.

Findings

Solutions form rational curves after analytic continuation

Differential invariant characterizes integrability

Infinite dimensional family of integrable systems

Abstract

The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.