Vector bundle automorphisms preserving Morse-Bott foliations

Let M be a smooth manifold and F a Morse-Bott foliation with a compact critical manifold Σ⊂M. Denote by D(F) 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(F) 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(F) consisting of diffeomorphisms fixed near Σ. Examples of computations of homotopy types of groups D(F) for such foliations are also presented.