  Let $\Phi$ be a flow on a smooth, compact, finite-dimensional manifold $M$.
Consider the subsets $E(\Phi)$ and $D(\Phi)$ of $C^{\infty}(M,M)$ consisting of
smoothh mappings and diffeomorphisms (respectively) of $M$ preserving the
foliation of the flow $\Phi$. Let also $E_{0}(\Phi)$ and $D_{0}(\Phi)$ be the
identity path components of $E(\Phi)$ and $D(\Phi)$ with compact-open topology.
We prove that under mild conditions on fixed points of $\Phi$ the inclusion
$D_{0}(\Phi) \subset E_{0}(\Phi)$ is a homotopy equivalence and these spaces
are either contractible or homotopically equivalent to the circle.
