Geometry of orbits of vector fields

In this paper, we study geometric and topological properties of vector fields on Riemannian manifolds of constant and nonnegative curvature, including Killing vector fields. We construct a completely integrable family of vector fields such that its orbits form a foliation whose set of singular fibers consists of two circles and regular fibers are two-dimensional tori. The solenoidal character of Killing vector fields on three-dimensional Euclidean space is also proved.