Topology of Ambient 3-Manifolds of Non-Singular Flows with Twisted Saddle Orbit

In the present paper, nonsingular Morse – Smale flows on closed orientable 3-manifolds are considered under the assumption that among the periodic orbits of the flow there is only one saddle and that it is twisted. An exhaustive description of the topology of such manifolds is obtained. Namely, it is established that any manifold admitting such flows is either a lens space or a connected sum of a lens space with a projective space, or Seifert manifolds with a base homeomorphic to a sphere and three singular fibers. Since the latter are prime manifolds, the result obtained refutes the claim that, among prime manifolds, the flows considered admit only lens spaces.