  Chazy studied a family of homogeneous third-order autonomous differential
equations. They are those, within a certain class, admitting exclusively
single-valued solutions. Each one of these equations yields a polynomial vector
field in complex three-dimensional space. For almost all of these these vector
fields, the Zariski closure of a generic orbit yields an affine surface endowed
with a holomorphic vector field that has exclusively single-valued solutions.
We classify these surfaces and relate this classification to recent results of
Rebelo and the author.
