  We obtain a theory of stratified Sternberg spaces thereby extending the
theory of cotangent bundle reduction for free actions to the singular case
where the action on the base manifold consists of only one orbit type. We find
that the symplectic reduced spaces are stratified topological fiber bundles
over the cotangent bundle of the orbit space. We also obtain a Poisson
stratification of the Sternberg space. To construct the singular Poisson
Sternberg space we develop an appropriate theory of singular connections for
proper group actions on a single orbit type manifold including a theory of
holonomy extending the usual Ambrose-Singer theorem for principal bundles.
