  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.
