Bianchi-IX, Darboux-Halphen and Chazy-Ramanujan

Bianchi-IX four metrics are $SU(2)$ invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler Top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux-Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux-Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang-Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux-Halphen system and occurring in the study of Bianchi IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.