  We study isometric circle actions on 7 dimensional positively curved
Eschenburg spaces which are almost free, thus giving rise to orbifold
fibrations of these spaces. This shows in particular that every known example
of compact manifolds with positive sectional curvature is the total space of an
orbifold fibration. We then study in detail the geometry of the singular locus
and how to minimize it. As a consequence we obtain, for example, a six
dimensional compact positively curved orbifold with only two singular points
with orbifold groups Z_2.
