The paper is a continuation of the authors et al.s work in the first half of the year 2021. It has classified a special class of 7-dimensional real solvable Lie algebras such that the nilradical of each from them is well-known 5-dimensional nilpotent Lie algebra in that work. In this paper, we will consider exponential, connected and simply connected Lie groups which are corresponding to these Lie algebras. Namely, we will describe the geometry of generic (i.e. 6-dimensional) orbits in coadjoint representation of considered Lie groups. Next, we will prove that for each considered group, the family of generic coadjoint orbits forms a measurable foliation in the sense of Connes and give the topological classification of these foliations.
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