TITLE: Are manifolds admitting a circle foliation covered by manifolds with a (non-trivial) circle action?
QUESTION [13 upvotes]: More precisely, is there a criterion that decides the above question?
I am particularly interested in the smooth setting: is a smooth manifold with a smooth regular foliation by circles covered by a smooth manifold with a smooth non-trivial circle action?
I am mostly interested in finite covers.

REPLY [11 votes]: Dennis Sullivan constructed a non-vanishing Lispschitz vector field on a closed $5$-manifold such that all orbits are periodic but they amazingly have unbounded lengths (!). An addendum by Nicholas Kuiper shows that the flow can be chosen smooth, and another by William Thurston gives another construction which is real analytic.