Vector bundle automorphisms preserving Morse-Bott foliations

Let M be a smooth manifold and $F$ a Morse-Bott foliation with a compact critical manifold $\Sigma \subset M$. Denote by $D\left(F\right)$ the group of diffeomorphisms of M leaving invariant each leaf of $F$. Under certain assumptions on $F$ it is shown that the computation of the homotopy type of $D\left(F\right)$ reduces to three rather independent groups: the group of diffeomorphisms of Σ, the group of vector bundle automorphisms of some regular neighborhood of Σ, and the subgroup of $D\left(F\right)$ consisting of diffeomorphisms fixed near Σ. Examples of computations of homotopy types of groups $D\left(F\right)$ for such foliations are also presented.