The Generalized Chazy Equation from the Self-Duality Equations

It is shown that classically known generalizations of the Chazy equation and Darboux–Halphen system are reductions of the self-dual Yang–Mills (SDYM) equations with an infinite-dimensional gauge algebra. The general ninth-order Darboux–Halphen system is reduced to a Schwarzian equation which governs conformal mappings of regions with piecewise circular sides. The generalized Chazy equation is shown to correspond to special mappings where either the triangles are equiangular or two of the angles are π/3.