The DH5 system and the Chazy and Ramamani equations

This article considers a class of nonlinear ordinary differential equations (ODEs) whose general solutions possess a natural boundary in the complex plane and admit special solutions that are automorphic under the action of subgroups of the modular group P ${\text{SL}}_2(\mathbb{Z})$. Specifically, a $3\times 3$ matrix valued system known as the Darboux-Halphen 9 (DH9) system and its fifth-order reduction to the DH5 system are discussed. These systems arise as reductions of the self-dual Yang-Mills equations of mathematical physics. It is shown that the equations discovered by J. Chazy (1908) and V. Ramamani (1970) are embedded in the DH5 system.