Local inverses of shift maps along orbits of flows

Let $M$ be a smooth manifold and $F$ be a vector field on $M$. My article ["Smooth shifts along trajectories of flows", Topol. Appl. 130 (2003) 183-204, arXiv:math/0106199] concerning the homotopy types of the group of diffeomorphisms preserving orbits of $F$ contains two errors. They imply that the principal statement of that paper holds under additional assumptions on $F$. Unfortunately this result was essentially used in another paper of mine ["Homotopy types of stabilizers and orbits of Morse functions on surfaces" Ann. Glob. Anal. Geom., 29 no. 3, (2006) 241-285, arXiv:math/0310067]. The aim of this article is to expose the results of the first paper in a right way, extend them to a larger class of flows with degenerate singularities, and show that the results of the second paper remain true.