  In this paper we consider non-singular Morse-Smale flows on closed orientable
3-manifolds, under the assumption that among the periodic orbits of the flow
there is only one saddle orbit and it is twisted. It is found 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 base sphere and three
special layers. A complete topological classification of the described flows is
obtained and the number of their equivalence classes on each admissible
manifold is calculated.
