Foliations Formed by Generic Coadjoint Orbits of Lie Groups Corresponding to a Class Seven-Dimensional Solvable Lie Algebras

We consider all connected and simply connected 7-dimensional Lie groups whose Lie algebras have nilradical ${\text{g}}_{5,2}=\text{s}\{X_1,X_2,X_3,X_4,X_5\text{\hspace{0.17em}⁣}:[X_1,X_2]=X_4,[X_1,X_3]=X_5\}$ of Dixmier. First, we give a geometric description of the maximal-dimensional orbits in the coadjoint representation of all considered Lie groups. Next, we prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. Finally, the topological classification of all these foliations is also provided.