Let $M$ be compact manifold and $\Phi$ be a smooth flow on $M$. Consider a map $\lambda:C^{\infty}(M) \to C^{\infty}(M,M)$ defined by the formula $\lambda(f)(x) = \Phi(x,f(x))$, where $f \in C^{\infty}(M)$ and $x\in M$. Note that each map $M \to M$ of a form $\lambda(f)$ preserves trajectories of $\Phi$. Let $\mathrm{Inv}\Phi \subset \mathrm{Diff} M$ be a subgroup consisting of diffeomorphisms which preserve each trajectory of $\Phi$ with its orientation. We introduce a large class $\mathcal{F}(M)$ of flows, which are "regular" extensions of linear flows near each of its fixed points and prove that for each flow $\Phi \in \mathcal{F}(M)$ a unity component of $\mathrm{Inv}\Phi$ is contractible in strong Whitney topology.
The results obtained are applied to homotopy groups study of the stabilizers and orbits of Morse functions on surfaces under the action of diffeomorphisms group.
