First integrals of a generalized Darboux-Halphen system.

A third-order system of nonlinear, ordinary differential equations depending on 3 arbitrary parameters is analyzed. The system arises in the study of SU(2)-invariant hypercomplex manifolds and is a dimensional reduction of the self-dual Yang-Mills equation. The general solution, first integrals and the Nambu-Poisson structure of the system are explicitly derived. It is shown that the first integrals are multi-valued on the phase space even though the general solution of the system is single-valued for special choices of parameters.