  Generalized Darboux-Halphen (gDH) systems, which form a versatile class of
three-dimensional homogeneous quadratic differential systems (HQDS's), are
introduced. They generalize the Darboux-Halphen (DH) systems considered by
other authors, in that any non-DH gDH system is affinely but not projectively
covariant. It is shown that the gDH class supports a rich collection of
rational solution-preserving maps: morphisms that transform one gDH system to
another. The proof relies on a bijection between (i) the solutions with
noncoincident components of any `proper' gDH system, and (ii) the solutions of
a generalized Schwarzian equation (gSE) associated to it, which generalizes the
Schwarzian equation (SE) familiar from the conformal mapping of hyperbolic
triangles. The gSE can be integrated parametrically in terms of the solutions
of a Papperitz equation, which is a generalized Gauss hypergeometric equation.
Ultimately, the rational gDH morphisms come from hypergeometric
transformations. A complete classification of proper non-DH gDH systems with
the Painleve property (PP) is also carried out, showing how some are related by
rational morphisms. The classification follows from that of non-SE gSE's with
the PP, due to Garnier and Carton-LeBrun. As examples, several non-DH gDH
systems with the PP are integrated explicitly in terms of elementary and
elliptic functions.
