# Foliations formed by generic coadjoint orbits of Lie groups corresponding to
a class 7-dimensional solvable Lie algebras

Tuyen T. M. Nguyen<EMAIL_ADDRESS>Faculty of Mathematics and Computer
Science, University of Science, VNU Ho Chi Minh City, Vietnam  Faculty of
Mathematics and Computer Science Teacher Education, Dong Thap University, Dong
Thap Province, Vietnam  Vu A. Le<EMAIL_ADDRESS>Department of Economic
Mathematics, University of Economics and Law, Vietnam National University – Ho
Chi Minh City, Vietnam  Tuan A. Nguyen<EMAIL_ADDRESS>Department
of Primary Education, Ho Chi Minh City University of Education, Vietnam

###### Abstract

We consider all connected and simply connected 7-dimensional Lie groups whose
Lie algebras have nilradical
$\mathfrak{g}_{5,2}=\operatorname{span}\\{X_{1},X_{2},X_{3},X_{4},X_{5}\colon[X_{1},X_{2}]=X_{4},[X_{1},X_{3}]=X_{5}\\}$
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.

###### keywords:

K-orbit , Lie algebra , Lie group , measurable foliation.

###### MSC:

[2010] 53C12 , 17B08 , 22E27 , 57R30 , 17B30 , 22E45.

††journal: Topology and its Applications

##  1 Introduction

The study of foliations on manifolds has a long history in mathematics. Since
the works of Ehresmann and Reeb [4] in 1944, Ehresmann [5] in 1951, and Reeb
[12] in 1952, the foliations on manifolds have enjoyed rapid development.
Nowadays, they become the focus of a great deal of research activity (see
[9]). In general, the leaf space of a foliation with quotient topology is a
fairly intractable topological space. To improve upon the shortcoming, Connes
[1] proposed the notion of measurable foliations in 1982 and associated each
such foliation $(V,\mathcal{F})$ with a C*-algebra $C^{*}(V,\mathcal{F})$. It
represents the leaf space $V/\mathcal{F}$ in the following sense: when the
foliation comes from a fibration (with connected fibers) $p:V\to M$ then
$C^{*}(V,\mathcal{F})$ is isomorphic to $C_{0}(M)\otimes\mathcal{K}$, where
$C_{0}(M)$ is the algebra of continuous complex-valued functions defined on
$M$ vanishing at infinity and $\mathcal{K}$ denotes the C*-algebra of compact
operators on an (infinite-dimensional) separable Hilbert space. During the
last few decades, these concepts of Connes have become important tools of non-
commutative differential geometry and have attracted much attention from
mathematicians around the world.

In 1962, Kirillov [8, Section 15] invented the method of orbits that quickly
became the most important one in the representation theory of Lie groups and
Lie algebras. The key to Kirillov’s method of orbits is generic (in a certain
sense) orbits in the coadjoint representation (K-orbits, for short) of Lie
groups. Hence, the problem of describing the geometry of (generic) K-orbits of
each Lie group is very important to study. A concept closely related to
Kirillov’s method of orbits is the so-called _MD-group_ proposed by Diep [2]
in 1980. An MD-group, in terms of Diep [2, Section 4.1], is a finite-
dimensional solvable real Lie group whose K-orbits are of dimension zero or
maximal dimension. It is very remarkable that the family of maximal-
dimensional K-orbits of an MD-group $G$ forms a so-called measurable foliation
in the sense of Connes [1] which is called MD-foliation associated with $G$.

Thus, the nature of the problem concerning MD-groups, MD-foliations, and their
Connes’ C*-algebras is an interesting combination of Kirillov’s method of
orbits with Connes’ method in non-commutative geometry. Therefore, this
problem is worth studying. During the period from 1990 to now, there have been
much research works on MD-groups and MD-foliations, typically working [18, 19,
20], [21] and [22, 23, 24], … Up today, the problem of classifying MD-groups
and MD-foliations still remains open.

Naturally, a problem arises as follows: For a solvable Lie group that is not
an MD-group, does the family of its maximal-dimensional K-orbits have the same
properties as the K-orbits of MD-groups? We hope that the answer is positive.
Namely, we hope that some nice properties of MD-groups can be generalized for
a larger class of these solvable Lie groups.

According to Dixmier [3, Proposition 1], the class of 5-dimensional real or
complex nilpotent Lie algebras consists of nine algebras which are denoted by
$(\mathfrak{g}_{1})^{5}$, $(\mathfrak{g}_{1})^{2}\oplus\mathfrak{g}_{3}$,
$\mathfrak{g}_{1}\oplus\mathfrak{g}_{4}$, $\mathfrak{g}_{5,1}$,
$\mathfrak{g}_{5,2}$, $\mathfrak{g}_{5,3}$, $\mathfrak{g}_{5,4}$,
$\mathfrak{g}_{5,5}$, $\mathfrak{g}_{5,6}$. Recently, we have classified
7-dimensional indecomposable solvable Lie algebras with nilradical
$(\mathfrak{g}_{1})^{2}\oplus\mathfrak{g}_{3},\mathfrak{g}_{1}\oplus\mathfrak{g}_{4},\mathfrak{g}_{5,2},\mathfrak{g}_{5,4}$
in [25]. Combining these results with those of Ndogmo and Winternitz [10],
Rubin and Winternitz [13], Šnobl and Karásek [14], Šnobl and Winternitz [15,
16], Gong [6], Parry [11], Hindeleh and Thompson [7], we achieve a full
classification of 7-dimensional indecomposable solvable Lie algebras. As a
continuation of [25], we combine the idea of Kirillov’s method of orbits with
Connes’ method in non-commutative geometry to consider all Lie groups
corresponding to Lie algebras classified in [25]. As stated above, we hope
that some nice properties of MD-groups can be generalized for a larger class
of these solvable Lie groups. Namely, we started to study Lie groups
corresponding to Lie algebras in Table LABEL:tab1 which are not too
complicated but also not trivial among the groups classified in [25]. In fact,
we have solved quite completely in [17] the similar problem of MD-groups for
five families of Lie groups corresponding to Lie algebras $\mathcal{G}_{2}$,
$\mathcal{G}_{3}$, $\mathcal{G}^{00}_{4}$, $\mathcal{G}_{9}$,
$\mathcal{G}^{\lambda}_{10}$ in Table LABEL:tab1.

Table 1: 7-dimensional solvable Lie algebras with nilradical $\mathfrak{g}_{5,2}$ No. |  Triplets $(a_{X},a_{Y},[X,Y])$ |  Conditions
---|---|---
$\mathcal{G}_{1}^{\lambda}$ |  $(1,-1,0,0,1)$, $(0,0,1,0,1)$, $\lambda X_{4}$ |  $\lambda\in\\{0,1\\}$
$\mathcal{G}_{2}$ |  $(1,0,0,1,1)$, $(0,0,1,0,1)$ |
$\mathcal{G}_{3}$ |  $(0,1,0,1,0)$, $(0,0,1,0,1)$ |
$\mathcal{G}^{\lambda_{1}\lambda_{2}}_{4}$ |  $(1,0,\lambda_{1},1,1+\lambda_{1})$, $(0,1,\lambda_{2},1,\lambda_{2})$ |  $(\lambda_{1},\lambda_{2})\neq(-1,0)$, $\lambda_{1}+1\neq\lambda_{2}$
$\mathcal{G}_{5}$ |  $(0,0,1,0,1)$, $(1,1,0,2,1)+E_{12}$ |
$\mathcal{G}^{\lambda}_{6}$ |  $(1,1,\lambda,2,1+\lambda)$, $(0,0,1,0,1)+E_{12}$ |  $\lambda\in\mathbb{R}$
$\mathcal{G}_{7}$ |  $(0,1,1,1,1)$, $(1,1,0,2,1)+E_{25}$ |
$\mathcal{G}^{\lambda}_{8}$ |  $(1,1+\lambda,\lambda,2+\lambda,1+\lambda)$, $(0,1,1,1,1)+E_{25}$ |  $\lambda\in\mathbb{R}$
$\mathcal{G}_{9}$ |  $(0,0,1,0,1)$, $(0,1,0,1,0)+E_{35}$ |
$\mathcal{G}^{\lambda}_{10}$ |  $(0,1,\lambda,1,\lambda)$, $(0,0,1,0,1)+E_{35}$ |  $\lambda\in\mathbb{R}$
$\mathcal{G}_{11}$ |  $(0,1,1,1,1)$, $(1,0,0,1,1)+E_{23}+E_{45}$ |
$\mathcal{G}^{\lambda}_{12}$ |  $(1,\lambda,\lambda,1+\lambda,1+\lambda)$, $(0,1,1,1,1)+E_{23}+E_{45}$ |  $\lambda\in\mathbb{R}\setminus\\{-1\\}$
$\mathcal{G}_{13}^{\lambda}$ |  $(0,1,1,1,1)$, $(\lambda,S_{01},S_{\lambda 1})$ |  $\lambda\geq 0$
$\mathcal{G}_{14}^{\lambda_{1}\lambda_{2}}$ |  $(1,\lambda_{1},\lambda_{1},1+\lambda_{1},1+\lambda_{1})$, $(0,S_{\lambda_{2}1},S_{\lambda_{2}1})$ |  $\lambda_{2}\geq 0,\lambda_{1}\neq-1$
$\mathcal{G}_{15}$ |  $(0,S_{01},S_{01})$, $(0,1,1,1,1)+E_{25}-E_{34}$ |
$\mathcal{G}_{16}^{\lambda}$ |  $(0,S_{01},S_{01})+E_{25}$, $(0,1,1,1,1)+\lambda(E_{25}-E_{34})$ |  $\lambda\geq 0$

Here, as usual, we use the convention that $[X,Y]$ disappears if $[X,Y]=0$; we
denote by $(a_{1},\dots,a_{5})$ the diagonal $(5\times 5)$-matrix
diag$(a_{1},\dots,a_{5})$, $E_{ij}$ is the $(5\times 5)$-matrix whose only
non-zero entry is 1 in row $i$, column $j$ ($1\leq i,j\leq 5$) and
$S_{ab}:=\begin{bmatrix}a&b\\\ -b&a\end{bmatrix}$.

In this paper, we continue to study the similar problem of MD-groups for all
Lie groups corresponding to the remaining Lie algebras in Table LABEL:tab1.
The main results of the paper are as follows. First, we describe the geometric
pictures of maximal-dimensional K-orbits of the considered Lie groups. Next,
we prove that the families of all generic maximal-dimensional K-orbits of the
Lie groups corresponding to considered Lie algebras in Table LABEL:tab1 form
measurable foliations (in the sense of Connes [1]). Finally, we give the
topological classification of these foliations.

The paper is organized into four sections, including this introduction.
Section 2 describes the geometry of generic K-orbits of considered Lie groups.
Section 3 is devoted to dealing with foliations formed by generic K-orbits. We
also give some concluding remarks in the last section.

##  2 Generic K-orbits of considered Lie groups

In [17, Section 2], we recalled some notions and well-known results about
adjoint and coadjoint representations, foliations and measurable foliations
which will be used later. For more details, we refer to the book of Kirillov
[8, $\S$15 and $\S$6]. Moreover, by [25], we have sixteen families of
connected and simply connected (real solvable) Lie groups corresponding to the
indecomposable Lie algebras given in Table LABEL:tab1. Note that throughout
this section the condition for the parameters
$\lambda,\lambda_{1},\lambda_{2}$ in each Lie group G is the same as the
condition for the parameters in each algebra $\mathcal{G}=\mathrm{Lie}(G)$ in
Table LABEL:tab1 if nothing more is said. In this section, we shall continue
using the notations as in [17]. Recall that the picture of maximal-dimensional
K-orbits of Lie groups
$G\in\\{G_{2},G_{3},G_{4}^{00},G_{9},G_{10}^{\lambda}\\}$ described in [17,
Theorem 19]. Here, we will describe the picture of maximal-dimensional
K-orbits of the remaining Lie groups.

For convenience, we put $\mathcal{A}$ be sixteen connected and simply
connected Lie groups corresponding to sixteen Lie algebras listed in Table
LABEL:tab1. For $G\in\mathcal{A}$, denote by $\mathcal{G}^{*}$ the dual space
of the Lie algebra $\mathcal{G}=\mathrm{Lie}(G)$ of $G$. Clearly, we can
identify $\mathcal{G}^{*}\equiv\mathbb{R}^{7}$ by fixing in it the basis
$(X_{1}^{*},\dots,X_{5}^{*},X^{*},Y^{*})$ which is the dual of the basis
$(X_{1},\dots,X_{5},X,Y)$ of $\mathcal{G}$. Let
$F=\alpha_{1}X_{1}^{*}+\dots+\alpha_{5}X_{5}^{*}+\alpha X^{*}+\beta
Y^{*}\equiv(\alpha_{1},\dots,\alpha_{5},\alpha,\beta)$ be an arbitrary element
of $\mathcal{G}^{*}\equiv\mathbb{R}^{7}$. The notation $\Omega_{F}$ will be
used to denote the K-orbit of $G$ containing $F$. By [17, Proposition 17,
Corollary 7], $G$ is exponential (then $\Omega_{F}=\Omega_{F}(\mathcal{G})$)
if it belongs to $\mathcal{A}_{1}:=\big{\\{}G^{\lambda}_{1}$, $G_{2}$,
$G_{3}$, $G_{4}^{\lambda_{1}\lambda_{2}}$, $G_{5}$, $G^{\lambda}_{6}$,
$G_{7}$, $G^{\lambda}_{8}$, $G_{9}$, $G^{\lambda}_{10}$, $G_{11}$,
$G_{12}^{\lambda}\big{\\}}$, and not exponential otherwise:
$G\in\mathcal{A}_{2}:=\left\\{G_{13}^{\lambda},G_{14}^{\lambda_{1}\lambda_{2}},G_{15},G_{16}^{\lambda}\right\\}$.
Of course, $\mathcal{A}=\mathcal{A}_{1}\sqcup\mathcal{A}_{2}$. Besides, we
have the following proposition.

######  Proposition 2.1 (see [8]).

Assume that $G$ is connected. Then $\Omega_{F}(\mathcal{G})=\Omega_{F}$ for
all $F\in\mathcal{G}^{*}$ if the family
$\left\\{\Omega_{F}(\mathcal{G})\right\\}_{F\in\mathcal{G}^{*}}$ forms a
partition of $\mathcal{G}^{*}$ and all
$\Omega_{F^{\prime}}(\mathcal{G}),F^{\prime}\in\Omega_{F}$ are either closed
or open (relatively) in $\Omega_{F},F\in\mathcal{G}^{*}$.

Therefore, we also have $\Omega_{F}=\Omega_{F}(\mathcal{G})$ for
$G\in\mathcal{A}_{2}$. That means $\Omega_{F}=\Omega_{F}(\mathcal{G})$ for all
$G\in\mathcal{A}$.

The geometric picture of the K-orbits of considered Lie groups is given as
follows.

######  Theorem 1.

Assume that $G\in\mathcal{A}$. Denote by
$(\alpha_{1},\dots,\alpha_{5},\alpha,\beta)$ the coordinate of
$F\in\mathcal{G}^{*}\equiv\mathbb{R}^{7}$ with respect to the basis
$(X_{1}^{*},\dots,X_{5}^{*},X^{*},Y^{*})$ which is dual one of the fixed basis
$(X_{1},\dots,X_{5},X,Y)$ of $\mathcal{G}$. Then, the maximal dimension of
K-orbits of $G$ is exactly six, and the picture of six-dimensional K-orbits of
$G$ is given in Table LABEL:tab2.

###### Proof.

The proof will take place through three steps as follows.

  * 1.

Step 1: For every pair $X,Y\in\mathcal{G}$ and
$F(\alpha_{1},\dots,\alpha_{5},\alpha,\beta)\in\mathcal{G}^{*}$, we need to
determine the matrix of the bilinear form $B_{F}(X,Y):=\langle
F,[X,Y]\rangle$.

  * 2.

Step 2: Find necessary and sufficient conditions for
$\operatorname{rank}B_{F}=6$ (the maximal dimension of K-orbits of $G$ is
exactly six).

  * 3.

Step 3: Describe explicitly every six-dimensional K-orbits of $G$, i.e.
describe
$\Omega_{F}\equiv\Omega_{F}(\mathcal{G})=\\{F_{U}\,|\,U\in\mathcal{G}\\}\subset\mathcal{G}^{*}\equiv\mathbb{R}^{7}$
where

$F_{U}=\sum_{i=1}^{5}x^{*}_{i}X^{*}_{i}+x^{*}X^{*}+y^{*}Y^{*}\equiv(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}$

is defined by $\left\langle F_{U},T\right\rangle=\left\langle
F,\exp(\operatorname{ad}_{U})T\right\rangle$ for all $T\in\mathcal{G}.$

First of all, the assertion of Theorem 1 for
$G\in\left\\{G_{2},G_{3},G_{9},G^{\lambda}_{10}\right\\}$ had been proved in
[17]. Below we will give detailed calculations for three groups
$G_{4}^{\lambda_{1}\lambda_{2}}$, $G_{12}^{\lambda}$ and $G_{13}^{\lambda}$
which do not belong to the family of groups considered in [17].

Let $G=G^{\lambda_{1}\lambda_{2}}_{4}$. In Step 1, by direct computations, the
matrix of the bilinear form $B_{F}$ is as follows

$B_{F}=\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&-\alpha_{1}&0\\\
-\alpha_{4}&0&0&0&0&0&-\alpha_{2}\\\
-\alpha_{5}&0&0&0&0&-\lambda_{1}\alpha_{3}&-\lambda_{2}\alpha_{3}\\\
0&0&0&0&0&-\alpha_{4}&-\alpha_{4}\\\
0&0&0&0&0&-(1+\lambda_{1})\alpha_{5}&-\lambda_{2}\alpha_{5}\\\
\alpha_{1}&0&\lambda_{1}\alpha_{3}&\alpha_{4}&(1+\lambda_{1})\alpha_{5}&0&0\\\
0&\alpha_{2}&\lambda_{2}\alpha_{3}&\alpha_{4}&\lambda_{2}\alpha_{5}&0&0\end{bmatrix}.$

For Step 2, it is obvious that the maximal rank of $B_{F}$ is six. Moreover

$\begin{array}[]{l l
l}\operatorname{rank}(B_{F})=6&\Leftrightarrow&\alpha_{4}=0\neq\alpha_{2}\alpha_{5}\quad\text{
or }\quad\alpha_{4}\neq 0\neq\alpha_{3}^{2}+\alpha_{5}^{2}.\end{array}$

In Step 3, we will describe the picture of maximal-dimensional K-orbits
$\Omega_{F}(\mathcal{G})=\\{F_{U}\,|\,U\in\mathcal{G}\\}\subset\mathcal{G}^{*}\equiv\mathbb{R}^{7}$
of $G$. To determine $F_{U}$ with
$U\equiv(x_{1},x_{2},x_{3},x_{4},x_{5},x,y)\in\mathcal{G}$, we use MAPLE to
determine $\exp(\operatorname{ad}_{U})$ with respect to the basis
$(X_{1},X_{2},X_{3},X_{4},X_{5},X,Y)$. We can find that

$\exp(\operatorname{ad}_{U})=\begin{bmatrix}e^{x}&0&0&0&0&-x_{1}\xi&0\\\
0&e^{y}&0&0&0&0&x_{2}\varepsilon\\\
0&0&e^{\lambda_{1}x+\lambda_{2}y}&0&0&\lambda_{1}x_{3}\zeta&\lambda_{2}x_{3}\zeta\\\
x_{2}e^{x}\varepsilon&x_{1}e^{y}\xi&0&e^{x+y}&0&b_{4}&d_{4}\\\
a_{4}&0&x_{1}e^{x\lambda_{1}+y\lambda_{2}}\xi&0&e^{\lambda_{1}x+\lambda_{2}y+x}&c_{4}&f_{4}\\\
0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix},$

where
$\xi=\frac{e^{x}-1}{x},\quad\varepsilon=\frac{1-e^{y}}{y},\quad\zeta=\frac{1-e^{x\lambda_{1}+y\lambda_{2}}}{x\lambda_{1}+y\lambda_{2}}$
and $a_{4},b_{4},c_{4},d_{4},f_{4}$ are expressions containing parameters
$x_{1},x_{3},x_{5},x,y$. By a direct computation, we get

$\begin{array}[]{l}x^{*}_{1}=\alpha_{1}e^{x}+\alpha_{4}\frac{x_{2}e^{x}(1-e^{y})}{y}+\alpha_{5}\frac{x_{3}e^{x}(1-e^{\lambda_{1}x+\lambda_{2}y})}{\lambda_{1}x+\lambda_{2}y}\\\
x^{*}_{2}=\alpha_{2}e^{y}+\alpha_{4}\frac{x_{1}e^{y}(e^{x}-1)}{x}\\\
x^{*}_{3}=\alpha_{3}e^{\lambda_{1}x+\lambda_{2}y}+\alpha_{5}\frac{x_{1}e^{\lambda_{1}x+\lambda_{2}y}(e^{x}-1)}{x}\\\
x^{*}_{4}=\alpha_{4}e^{x+y}\\\
x^{*}_{5}=\alpha_{5}e^{\lambda_{1}x+x+\lambda_{2}y}\\\
x^{*}=\alpha_{1}\frac{x_{1}(1-e^{x})}{x}+\alpha_{3}\lambda_{1}x_{3}\frac{1-e^{\lambda_{1}x+\lambda_{2}y}}{\lambda_{1}x+\lambda_{2}y}+\alpha_{4}p+\alpha_{5}q+\alpha\\\
y^{*}=\alpha_{2}x_{2}\frac{1-e^{y}}{y}+\alpha_{3}\lambda_{2}x_{3}\frac{1-e^{\lambda_{1}x+\lambda_{2}y}}{\lambda_{1}x+\lambda_{2}y}+\alpha_{4}r+\alpha_{5}s+\beta.\end{array}$

In order to describe all maximal-dimensional K-orbits of
$G=G^{\lambda_{1}\lambda_{2}}_{4}$, we only consider
$F(\alpha_{1},\dots,\alpha_{5},\alpha,\beta)\in\mathcal{G}^{*}$ with
$\alpha_{4}=0\neq\alpha_{2}\alpha_{5}$ or $\alpha_{4}\neq
0\neq\alpha_{3}^{2}+\alpha_{5}^{2}$, the remaining parameters are arbitrary.
There are three cases as follows.

  * 1.

The first case is $\alpha_{4}=0\neq\alpha_{2}\alpha_{5}$. Obviously, each of
$x^{*}_{1},x^{*}_{2},x^{*},y^{*}$ runs over line $\mathbb{R}$, while
$x^{*}_{4}\equiv 0$, and $x^{*}_{2},x^{*}_{5}\in\mathbb{R}$ satisfying
$\alpha_{2}x^{*}_{2}>0$ and $\alpha_{5}x^{*}_{5}>0$. Hence, $\Omega_{F}$ is a
part of hyperplane as follows:

$\Omega_{F}=\left\\{(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}\,\big{|}\,x^{*}_{4}=0,\alpha_{2}x^{*}_{2}>0,\alpha_{5}x^{*}_{5}>0\right\\}.$

  * 2.

The second case is $\alpha_{5}=0\neq\alpha_{3}\alpha_{4}$. Obviously, each of
$x^{*}_{1},x^{*}_{2},x^{*},y^{*}$ runs over line $\mathbb{R}$, while
$x^{*}_{5}\equiv 0$, and $x^{*}_{3},x^{*}_{4}\in\mathbb{R}$ satisfying
$\alpha_{3}x^{*}_{3}>0$ and $\alpha_{4}x^{*}_{4}>0$. For this reason,
$\Omega_{F}$ is a part of hyperplane as follows:

$\Omega_{F}=\left\\{(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}\,\big{|}\,x^{*}_{5}=0,\alpha_{3}x^{*}_{3}>0,\alpha_{4}x^{*}_{4}>0\right\\}.$

  * 3.

The final case is $\alpha_{4}\alpha_{5}\neq 0$. Clearly, each of
$x^{*}_{1},x^{*},y^{*}$ runs over line $\mathbb{R}$, while
$x^{*}_{4},x^{*}_{5}\in\mathbb{R}$ satisfying $\alpha_{4}x^{*}_{4}>0$ and
$\alpha_{5}x^{*}_{5}>0$. The coordinates
$x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5}$ satisfy the following equation

$\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}=\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{{\alpha_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{\alpha_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}.$

Thus, $\Omega_{F}$ is a part of hypersurface of degree two as follows:

$\displaystyle\Omega_{F}=\left\\{(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}\;\Bigg{|}\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}=\right.$
$\displaystyle\left.=\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{{\alpha_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{\alpha_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}},\;\alpha_{4}x^{*}_{4}>0,\,\alpha_{5}x^{*}_{5}>0\right\\}$

This completes the proof for $G=G^{\lambda_{1}\lambda_{2}}_{4}$.

Let $G=G^{\lambda}_{12}$. In Step 1, by direct computations, the matrix of the
bilinear form $B_{F}$ is as follows

$B_{F}=\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&-\alpha_{1}&0\\\
-\alpha_{4}&0&0&0&0&-\lambda\alpha_{2}&-(\alpha_{2}+\alpha_{3})\\\
-\alpha_{5}&0&0&0&0&-\lambda\alpha_{3}&-\alpha_{3}\\\
0&0&0&0&0&-(1+\lambda)\alpha_{4}&-(\alpha_{4}+\alpha_{5})\\\
0&0&0&0&0&-(1+\lambda)\alpha_{5}&-\alpha_{5}\\\
\alpha_{1}&\lambda\alpha_{2}&\lambda\alpha_{3}&(1+\lambda)\alpha_{4}&(1+\lambda)\alpha_{5}&0&0\\\
0&\alpha_{2}+\alpha_{3}&\alpha_{3}&\alpha_{4}+\alpha_{5}&\alpha_{5}&0&0\end{bmatrix}.$

For Step 2, it is obvious that the maximal rank of $B_{F}$ is six. Moreover

$\begin{array}[]{l l
l}\operatorname{rank}(B_{F})=6&\Leftrightarrow&\alpha_{3}\alpha_{4}\neq
0=\alpha_{5}\mbox{ or }\alpha_{5}\neq 0.\end{array}$

In Step 3, we also find that

$\exp(\operatorname{ad}_{U})=\begin{bmatrix}e^{x}&0&0&0&0&-x_{1}p&0\\\
0&e^{\lambda x+y}&0&0&0&-\lambda x_{2}r&-x_{2}r\\\ 0&ye^{\lambda
x+y}&e^{\lambda x+y}&0&0&b_{12}&f_{12}\\\ -x_{2}e^{x}r&x_{1}e^{\lambda
x+y}p&0&e^{\xi}&0&c_{12}&g_{12}\\\ a_{12}&x_{1}ye^{\lambda
x+y}p&x_{1}e^{\lambda x+y}p&ye^{\xi}&e^{\xi}&d_{12}&h_{12}\\\ 0&0&0&0&0&1&0\\\
0&0&0&0&0&0&1\end{bmatrix},$

where

$\begin{array}[]{l}\xi=(1+\lambda)x+y,\quad p=\dfrac{e^{x}-1}{x},\quad
r=\dfrac{e^{\lambda x+y}-1}{\lambda x+y}\\\ a_{12},b_{12},f_{12}:\text{
expressions containing parameters }x_{2},x_{3},x,y\\\
c_{12},d_{12},g_{12},h_{12}:\text{ expressions containing parameters
}x_{1},x_{2},x_{3},x_{4},x_{5},x,y.\end{array}$

By a direct computation, we get

$\begin{array}[]{l}x^{*}_{1}=\alpha_{1}e^{x}-\alpha_{4}x_{2}e^{x}r+\alpha_{5}a_{12}\\\
x^{*}_{2}=\alpha_{2}e^{\lambda x+y}+\alpha_{3}ye^{\lambda
x+y}+\alpha_{4}x_{1}e^{\lambda x+y}p+\alpha_{5}x_{1}ye^{\lambda x+y}p\\\
x^{*}_{3}=\alpha_{3}e^{\lambda x+y}+\alpha_{5}x_{1}e^{\lambda x+y}p\\\
x^{*}_{4}=\alpha_{4}e^{\xi}+\alpha_{5}ye^{\xi}\\\
x^{*}_{5}=\alpha_{5}e^{\xi}\\\ x^{*}=-\alpha_{1}x_{1}p-\alpha_{2}\lambda
x_{2}r+\alpha_{3}b_{12}+\alpha_{4}c_{12}+\alpha_{5}d_{12}+\alpha\\\
y^{*}=-\alpha_{2}x_{2}r+\alpha_{3}f_{12}+\alpha_{4}g_{12}+\alpha_{5}h_{12}+\beta.\end{array}$

To describe all maximal dimensional K-orbits of $G=G^{\lambda}_{12}$, we only
consider $F(\alpha_{1},\dots,\alpha_{5},\alpha,\beta)\in\mathcal{G}^{*}$ with
$\alpha_{3}\alpha_{4}\neq 0=\alpha_{5}$ or $\alpha_{5}\neq 0$, the remaining
parameters are arbitrary.

  * 1.

The second case, $\alpha_{3}\alpha_{4}\neq 0=\alpha_{5}$. Obviously, each of
$x^{*}_{1},x^{*}_{2},x^{*},y^{*}$ runs over line ${\mathbb{R}}$, while
$x^{*}_{5}\equiv 0$ and
$x^{*}_{3},x^{*}_{4}\in{\mathbb{R}};\,\alpha_{3}x^{*}_{3}>0,\,\alpha_{4}x^{*}_{4}>0$.
For this reason, we get $\Omega_{F}$ is a part of hyperplane as follows:

$\begin{array}[]{r}\Omega_{F}=\big{\\{}(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}:x^{*}_{5}=0,\,\alpha_{3}x^{*}_{3}>0,\,\alpha_{4}x^{*}_{4}>0\big{\\}}.\end{array}$

  * 2.

The final case, $\alpha_{5}\neq 0$. Clearly, each of $x^{*}_{1},x^{*},y^{*}$
runs over line ${\mathbb{R}}$, while
$x^{*}_{5}\in{\mathbb{R}};\,\alpha_{5}x^{*}_{5}>0$. By an easy computation it
follows that the coordinates $x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5}$ are
satisfy the following equation

$\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{1}{{x^{*}_{5}}^{\frac{\lambda}{1+\lambda}}e^{\frac{x^{*}_{4}}{(1+\lambda)x^{*}_{5}}}}=\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{1}{\alpha_{5}^{\frac{\lambda}{1+\lambda}}e^{\frac{\alpha_{4}}{(1+\lambda)\alpha_{5}}}}\cdot$

Thus, we get $\Omega_{F}$ is a half of hypersurface of degree two as follows:

$\begin{array}[]{rl}\Omega_{F}=\Bigg{\\{}(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}:&\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{1}{{x^{*}_{5}}^{\frac{\lambda}{1+\lambda}}e^{\frac{x^{*}_{4}}{(1+\lambda)x^{*}_{5}}}}=\\\
&=\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{1}{\alpha_{5}^{\frac{\lambda}{1+\lambda}}e^{\frac{\alpha_{4}}{(1+\lambda)\alpha_{5}}}},\,\alpha_{5}x^{*}_{5}>0\Bigg{\\}}.\end{array}$

This completes the proof for $G=G^{\lambda}_{12}$.

Let $G=G^{\lambda}_{13}$. In Step 1, by direct computations, the matrix of the
bilinear form $B_{F}$ is as follows

$B_{F}=\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&0&-\lambda\alpha_{1}\\\
-\alpha_{4}&0&0&0&0&-\alpha_{2}&-\alpha_{3}\\\
-\alpha_{5}&0&0&0&0&-\alpha_{3}&\alpha_{2}\\\
0&0&0&0&0&-\alpha_{4}&-(\lambda\alpha_{4}+\alpha_{5})\\\
0&0&0&0&0&-\alpha_{5}&\alpha_{4}-\lambda\alpha_{5}\\\
0&\alpha_{2}&\alpha_{3}&\alpha_{4}&\alpha_{5}&0&0\\\
\lambda\alpha_{1}&\alpha_{3}&-\alpha_{2}&\lambda\alpha_{4}+\alpha_{5}&-(\alpha_{4}-\lambda\alpha_{5})&0&0\end{bmatrix}.$

For Step 2, it is obvious that the maximal rank of $B_{F}$ is six. Moreover

$\begin{array}[]{l l
l}\operatorname{rank}(B_{F})=6&\Leftrightarrow&\alpha^{2}_{4}+\alpha^{2}_{5}\neq
0.\end{array}$

In Step 3, we also find that

$\exp(\operatorname{ad}_{U})=\begin{bmatrix}e^{\lambda y}&0&0&0&0&0&h_{13}\\\
0&e^{x}\cos(y)&-e^{x}\sin(y)&0&0&c_{13}&m_{13}\\\
0&e^{x}\sin(y)&e^{x}\cos(y)&0&0&d_{13}&n_{13}\\\
a_{13}&x_{1}e^{x}\cos(y)&-x_{1}e^{x}\sin(y)&e^{\xi}\cos(y)&-e^{\xi}\sin(y)&f_{13}&k_{13}\\\
b_{13}&x_{1}e^{x}\sin(y)&x_{1}e^{x}\cos(y)&e^{\xi}\sin(y)&e^{\xi}\cos(y)&g_{13}&l_{13}\\\
0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix},$

where

$\begin{array}[]{l}\xi=x+\lambda y,\\\
a_{13},b_{13},c_{13},d_{13},m_{13},n_{13}\colon\text{expressions containing
parameters }x_{2},x_{3},x,y,\\\
f_{13},g_{13},k_{13},l_{13}\colon\text{expressions containing parameters
}x_{1},x_{2},x_{3},x_{4},x_{5},x,y,\\\ h_{13}\colon\text{an expression
containing parameters }x_{1},x,y.\end{array}$

By a direct computation, we get

$\begin{array}[]{l}x^{*}_{1}=\alpha_{1}e^{\lambda
y}+\alpha_{4}a_{13}+\alpha_{5}b_{13}\\\
x^{*}_{2}=\alpha_{2}e^{x}\cos(y)+\alpha_{3}e^{x}\sin(y)+\alpha_{4}x_{1}e^{x}\cos(y)+\alpha_{5}x_{1}e^{x}\sin(y)\\\
x^{*}_{3}=-\alpha_{2}e^{x}\sin(y)+\alpha_{3}e^{x}\cos(y)-\alpha_{4}x_{1}e^{x}\sin(y)+\alpha_{5}x_{1}e^{x}\cos(y)\\\
x^{*}_{4}=\alpha_{4}e^{x+\lambda y}\cos(y)+\alpha_{4}5e^{x+\lambda
y}\sin(y)\\\ x^{*}_{5}=-\alpha_{4}e^{x+\lambda
y}\sin(y)+\alpha_{4}5e^{x+\lambda y}\cos(y)\\\
x^{*}=\alpha_{2}c_{13}+\alpha_{3}d_{13}+\alpha_{4}f_{13}+\alpha_{5}g_{13}+\alpha\\\
y^{*}=\alpha_{1}h_{13}+\alpha_{2}m_{13}+\alpha_{3}n_{13}+\alpha_{4}k_{13}+\alpha_{5}l_{13}+\beta.\end{array}$

To describe all maximal-dimensional K-orbits of $G=G^{\lambda}_{13}$, we only
consider $F(\alpha_{1},\dots,\alpha_{5},\alpha,\beta)\in\mathcal{G}^{*}$ with
$\alpha_{4}^{2}+\alpha^{2}_{5}\neq 0$, the remaining parameters are arbitrary.
Clearly, $x^{*}_{1},x^{*},y^{*}$ run over line $\mathbb{R}$, while
$x^{*}_{4},x^{*}_{5}\in\mathbb{R};\,{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}\neq 0$.
The coordinates $x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5}$ satisfy the
following equation

$\frac{x^{*}_{2}x^{*}_{5}-x^{*}_{3}x^{*}_{4}}{{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}}e^{\lambda\arctan\frac{x^{*}_{4}}{x^{*}_{5}}}=\frac{\alpha_{2}\alpha_{5}-\alpha_{3}\alpha_{4}}{{\alpha_{4}}^{2}+{\alpha_{5}}^{2}}e^{\lambda\arctan\frac{\alpha_{4}}{\alpha_{5}}}\cdot$

Thus, $\Omega_{F}$ is a part of hypersurface of degree two as follows:

$\begin{array}[]{l}\Omega_{F}=\Bigg{\\{}(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}:\frac{x^{*}_{2}x^{*}_{5}-x^{*}_{3}x^{*}_{4}}{{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}}e^{\lambda\arctan\frac{x^{*}_{4}}{x^{*}_{5}}}=\frac{\alpha_{2}\alpha_{5}-\alpha_{3}\alpha_{4}}{{\alpha_{4}}^{2}+{\alpha_{5}}^{2}}e^{\lambda\arctan\frac{\alpha_{4}}{\alpha_{5}}},\,{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}\neq
0\Bigg{\\}}.\end{array}$

This completes the proof for $G=G^{\lambda}_{13}$.

The remaining cases can be proved similarly while the matrices of $B_{F}$ with
necessary and sufficient conditions for $\operatorname{rank}B_{F}=6$ and
$\exp(\operatorname{ad}_{U})$ are given in Tables LABEL:tab5.2 and
LABEL:tab5.3, respectively.

Table 2:  $B_{F}$ and Condition of $\operatorname{rank}(B_{F})=6$ No. |  $B_{F}$, $F(\alpha_{1},\dots,\alpha_{5},\alpha,\beta)\in\mathcal{G}^{*}$ |  Condition of $\operatorname{rank}(B_{F})=6$
---|---|---
$\mathcal{G}^{\lambda}_{1}$ |  $\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&-\alpha_{1}&0\\\ -\alpha_{4}&0&0&0&0&\alpha_{2}&0\\\ -\alpha_{5}&0&0&0&0&0&-\alpha_{3}\\\ 0&0&0&0&0&0&0\\\ 0&0&0&0&0&-\alpha_{5}&-\alpha_{5}\\\ \alpha_{1}&-\alpha_{2}&0&0&\alpha_{5}&0&\alpha_{4}\\\ 0&0&\alpha_{3}&0&\alpha_{5}&-\alpha_{4}&0\end{bmatrix}$ |  $\alpha_{5}\neq 0\neq\alpha_{2}^{2}+\alpha_{4}^{2}$
$\mathcal{G}_{5}$ |  $\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&0&-(\alpha_{1}+\alpha_{2})\\\ -\alpha_{4}&0&0&0&0&0&-\alpha_{2}\\\ -\alpha_{5}&0&0&0&0&-\alpha_{3}&0\\\ 0&0&0&0&0&0&-2\alpha_{4}\\\ 0&0&0&0&0&-\alpha_{5}&-\alpha_{5}\\\ 0&0&\alpha_{3}&0&\alpha_{5}&0&0\\\ \alpha_{1}+\alpha_{2}&\alpha_{2}&0&2\alpha_{4}&\alpha_{5}&0&0\end{bmatrix}$ |  $\begin{array}[]{l}\alpha_{4}=0\neq\alpha_{2}\alpha_{5}\\\ \mbox{ or }\alpha_{4}\neq 0\neq\alpha_{3}^{2}+\alpha_{5}^{2}\end{array}$
$\mathcal{G}^{\lambda}_{6}$ |  $\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&-\alpha_{1}&-\alpha_{2}\\\ -\alpha_{4}&0&0&0&0&-\alpha_{2}&0\\\ -\alpha_{5}&0&0&0&0&-\lambda\alpha_{3}&-\alpha_{3}\\\ 0&0&0&0&0&-2\alpha_{4}&0\\\ 0&0&0&0&0&-(1+\lambda)\alpha_{5}&-\alpha_{5}\\\ \alpha_{1}&\alpha_{2}&\lambda\alpha_{3}&2\alpha_{4}&(1+\lambda)\alpha_{5}&0&0\\\ \alpha_{2}&0&\alpha_{3}&0&\alpha_{5}&0&0\end{bmatrix}$ |  $\begin{array}[]{l}\alpha_{4}=0\neq\alpha_{2}\alpha_{5}\\\ \mbox{ or }\alpha_{4}\neq 0\neq\alpha_{3}^{2}+\alpha_{5}^{2}\end{array}$
$\mathcal{G}_{7}$ |  $\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&0&-\alpha_{1}\\\ -\alpha_{4}&0&0&0&0&-\alpha_{2}&-(\alpha_{2}+\alpha_{5})\\\ -\alpha_{5}&0&0&0&0&-\alpha_{3}&0\\\ 0&0&0&0&0&-\alpha_{4}&-2\alpha_{4}\\\ 0&0&0&0&0&-\alpha_{5}&-\alpha_{5}\\\ 0&\alpha_{2}&\alpha_{3}&\alpha_{4}&\alpha_{5}&0&0\\\ \alpha_{1}&\alpha_{2}+\alpha_{5}&0&2\alpha_{4}&\alpha_{5}&0&0\end{bmatrix}$ |  $\begin{array}[]{l}\alpha_{4}^{2}+\alpha^{2}_{5}\neq 0\\\ \mbox{ except to }\\\ \alpha_{3}=\alpha_{5}=0\neq\alpha_{4}\end{array}$
$\mathcal{G}^{\lambda}_{8}$ |  $\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&-\alpha_{1}&0\\\ -\alpha_{4}&0&0&0&0&-\gamma&-(\alpha_{2}+\alpha_{5})\\\ -\alpha_{5}&0&0&0&0&-\lambda\alpha_{3}&\alpha_{3}\\\ 0&0&0&0&0&-\delta&-\alpha_{4}\\\ 0&0&0&0&0&-\xi&-\alpha_{5}\\\ \alpha_{1}&\gamma&\lambda\alpha_{3}&\delta&\xi&0&0\\\ 0&\alpha_{2}+\alpha_{5}&\alpha_{3}&\alpha_{4}&\alpha_{5}&0&0\end{bmatrix}$ |  $\begin{array}[]{l}\alpha_{4}^{2}+\alpha^{2}_{5}\neq 0\\\ \mbox{ except to }\\\ \alpha_{3}=\alpha_{5}=0\neq\alpha_{4}\end{array}$
$\mathcal{G}_{11}$ |  $\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&0&-\alpha_{1}\\\ -\alpha_{4}&0&0&0&0&-\alpha_{2}&-\alpha_{3}\\\ -\alpha_{5}&0&0&0&0&-\alpha_{3}&0\\\ 0&0&0&0&0&-\alpha_{4}&-(\alpha_{4}+\alpha_{5})\\\ 0&0&0&0&0&-\alpha_{5}&-\alpha_{5}\\\ 0&\alpha_{2}&\alpha_{3}&\alpha_{4}&\alpha_{5}&0&0\\\ \alpha_{1}&\alpha_{3}&0&\alpha_{4}+\alpha_{5}&\alpha_{5}&0&0\end{bmatrix}$ |  $\alpha_{4}\neq 0\neq{\alpha_{3}}^{2}+{\alpha_{5}}^{2}$
$\mathcal{G}^{\lambda_{1},\lambda_{2}}_{14}$ |  $\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&-\alpha_{1}&0\\\ -\alpha_{4}&0&0&0&0&-\lambda_{1}\alpha_{2}&-\beta\\\ -\alpha_{5}&0&0&0&0&-\lambda_{1}\alpha_{3}&-\gamma\\\ 0&0&0&0&0&-\delta&-\xi\\\ 0&0&0&0&0&-\sigma&-\chi\\\ \alpha_{1}&\lambda_{1}\alpha_{2}&\lambda_{1}\alpha_{3}&\delta&\sigma&0&0\\\ 0&\beta&\gamma&\xi&\chi&0&0\end{bmatrix}$ |  $\alpha^{2}_{4}+\alpha^{2}_{5}\neq 0$
|  where $\begin{array}[]{ll}\beta=\lambda_{2}\alpha_{2}+\alpha_{3},&\gamma=-\alpha_{2}+\lambda_{2}\alpha_{3}\\\ \delta=(1+\lambda_{1})\alpha_{4},&\sigma=(1+\lambda_{1})\alpha_{5}\\\ \xi=\lambda_{2}\alpha_{4}+\alpha_{5},&\chi=-\alpha_{4}+\lambda_{2}\alpha_{5}\end{array}$ |
$\mathcal{G}_{15}$ |  $\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&0&0\\\ -\alpha_{4}&0&0&0&0&-\alpha_{3}&-\beta\\\ -\alpha_{5}&0&0&0&0&\alpha_{2}&-\gamma\\\ 0&0&0&0&0&-\alpha_{5}&-\alpha_{4}\\\ 0&0&0&0&0&\alpha_{4}&-\alpha_{5}\\\ 0&\alpha_{3}&-\alpha_{2}&\alpha_{5}&-\alpha_{4}&0&0\\\ 0&\beta&\gamma&\alpha_{4}&\alpha_{5}&0&0\end{bmatrix}$ |  $\alpha^{2}_{4}+\alpha^{2}_{5}\neq 0$
|  where $\beta=\alpha_{2}+\alpha_{5},\gamma=\alpha_{3}-\alpha_{4}$ |
$\mathcal{G}^{\lambda}_{16}$ |  $\begin{bmatrix}0&\alpha_{4}&\alpha_{5}&0&0&0&0\\\ -\alpha_{4}&0&0&0&0&-\beta&-\gamma\\\ -\alpha_{5}&0&0&0&0&\alpha_{2}&-\sigma\\\ 0&0&0&0&0&-\alpha_{5}&-\alpha_{4}\\\ 0&0&0&0&0&\alpha_{4}&-\alpha_{5}\\\ 0&\beta&-\alpha_{2}&\alpha_{5}&-\alpha_{4}&0&0\\\ 0&\gamma&\sigma&\alpha_{4}&\alpha_{5}&0&0\end{bmatrix}$ |  $\alpha^{2}_{4}+\alpha^{2}_{5}\neq 0$
|  where $\begin{array}[]{ll}\beta=\alpha_{3}+\alpha_{5},&\gamma=\alpha_{2}+\lambda\alpha_{5},\\\ \sigma=\alpha_{3}-\lambda\alpha_{4}&\end{array}$ |
Table 2:  (continued) Table 3:  $\exp(\operatorname{ad}_{U})$ of considered Lie groups No. |  $\exp(\operatorname{ad}_{U}),\;U=\sum_{i=1}^{5}x_{i}X_{i}+xX+yY\;(x_{i},x,y\in\mathbb{R})$
---|---
$G_{1}^{\lambda}$ |  $\begin{bmatrix}e^{x}&0&0&0&0&-x_{1}p&0\\\ 0&e^{-x}&0&0&0&x_{2}e^{-x}p&0\\\ 0&0&e^{y}&0&0&0&-x_{3}q\\\ -x_{2}p&x_{1}e^{-x}p&0&1&0&a_{1}&\lambda x\\\ -x_{3}e^{x}q&0&x_{1}e^{y}p&0&e^{x+y}&b_{1}&c_{1}\\\ 0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix}$
|  where $\begin{cases}p=\dfrac{e^{x}-1}{x},\quad q=\dfrac{e^{y}-1}{y}\\\
a_{1}:\text{ an expression containing parameters }x_{1},x_{2},x,y\\\
b_{1},c_{1}:\text{ expressions containing parameters
}x_{1},x_{3},x_{5},x,y.\end{cases}$
$G_{5}$ |  $\begin{bmatrix}e^{y}&0&0&0&0&0&-x_{1}q\\\ e^{y}y&e^{y}&0&0&0&0&c_{5}\\\ 0&0&e^{x}&0&0&-x_{3}p&0\\\ a_{5}&x_{1}e^{y}q&0&e^{2y}&0&0&d_{5}\\\ -x_{3}e^{y}p&0&x_{1}e^{x}q&0&e^{x+y}&b_{5}&f_{5}\\\ 0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix}$
|  where $\begin{cases}a_{5},c_{5}:\text{ expressions containing parameters
}x_{1},x_{2},x,y\\\ d_{5}:\text{ an expression containing parameters
}x_{1},x_{4},y\\\ b_{5},f_{5}:\text{ expressions containing parameters
}x_{1},x_{3},x_{5},x,y.\end{cases}$
$G_{6}^{\lambda}$ |  $\begin{bmatrix}e^{x}&0&0&0&0&-x_{1}p&0\\\ e^{x}y&e^{x}&0&0&0&b_{6}&-x_{1}p\\\ 0&0&e^{\lambda x+y}&0&0&-\lambda x_{3}r&-x_{3}r\\\ a_{6}&x_{1}e^{x}p&0&e^{2x}&0&c_{6}&f_{6}\\\ -x_{3}e^{x}r&0&x_{1}e^{\lambda x+y}p&0&e^{\xi}&d_{6}&g_{6}\\\ 0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix}$
|  where $\begin{cases}\xi=\lambda x+x+y,\quad r=\dfrac{e^{\lambda
x+y}-1}{\lambda x+y}\\\ a_{6},b_{6}:\text{ expressions containing parameters
}x_{1},x_{2},x,y\\\ c_{6}:\text{ an expression containing parameters
}x_{1},x_{4},x,y\\\ f_{6}:\text{ an expression containing parameters
}x_{1},x\\\ d_{6},g_{6}:\text{ expressions containing parameters
}x_{1},x_{3},x_{5},x,y.\end{cases}$
$G_{7}$ |  $\begin{bmatrix}e^{y}&0&0&0&0&0&-x_{1}q\\\ 0&e^{x+y}&0&0&0&b_{7}&b_{7}\\\ 0&0&e^{x}&0&0&-x_{3}p&0\\\ a_{7}&x_{1}e^{x+y}q&0&e^{x+2y}&0&c_{7}&f_{7}\\\ -x_{3}e^{y}p&ye^{x+y}&x_{1}e^{x}&0&e^{x+y}&d_{7}&g_{7}\\\ 0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix}$
|  where $\begin{cases}a_{7},b_{7}:\text{ expressions containing parameters
}x_{2},x,y\\\ c_{7},f_{7}:\text{ expressions containing parameters
}x_{1},x_{2},x_{4},x,y\\\ d_{7},g_{7}:\text{ expressions containing parameters
}x_{1},x_{2},x_{3},x_{5},x,y.\end{cases}$
$G_{8}^{\lambda}$ |  $\begin{bmatrix}e^{x}&0&0&0&0&-x_{1}p&0\\\ 0&e^{\xi}&0&0&0&-(1+\lambda)x_{2}s&-x_{2}s\\\ 0&0&e^{\lambda x+y}&0&0&-\lambda x_{3}r&-x_{3}r\\\ -x_{2}e^{x}s&x_{1}e^{\xi}p&0&e^{x+\xi}&0&a_{8}&c_{8}\\\ -x_{3}e^{x}r&ye^{\xi}&x_{1}e^{\lambda x+y}p&0&e^{\xi}&b_{8}&d_{8}\\\ 0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix}$
|  where $\begin{cases}s=\dfrac{e^{\lambda x+x+y}-1}{\lambda x+x+y}\\\
a_{8},c_{8}:\text{ expressions containing parameters }x_{1},x_{2},x_{4},x,y\\\
b_{8},d_{8}:\text{ expressions containing parameters
}x_{1},x_{2},x_{3},x_{5},x,y.\end{cases}$
$G_{11}$ |  $\begin{bmatrix}e^{y}&0&0&0&0&0&-x_{1}q\\\ 0&e^{x}&0&0&0&-x_{2}p&0\\\ 0&ye^{x}&e^{x}&0&0&b_{11}&-x_{2}p\\\ -x_{2}e^{y}p&x_{1}e^{x}q&0&e^{x+y}&0&c_{11}&f_{11}\\\ a_{11}&x_{1}e^{x}(e^{y}-1)&x_{1}e^{x}q&ye^{x+y}&e^{x+y}&d_{11}&g_{11}\\\ 0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix}$
|  where $\begin{cases}a_{11},b_{11}:\text{ expressions containing parameters
}x_{2},x_{3},x,y\\\ c_{11},f_{11}:\text{ expressions containing parameters
}x_{1},x_{2},x_{4},x,y\\\ d_{11},g_{11}:\text{ expressions containing
parameters }x_{1},x_{2},x_{3},x_{4},x_{5},x,y.\end{cases}$
$G_{14}^{\lambda_{1},\lambda_{2}}$ |  $\begin{bmatrix}e^{x}&0&0&0&0&c_{14}&0\\\ 0&e^{\zeta}\cos(y)&-e^{\zeta}\sin(y)&0&0&d_{14}&m_{14}\\\ 0&e^{\zeta}\sin(y)&e^{\zeta}\cos(y)&0&0&f_{14}&n_{14}\\\ a_{14}&x_{1}e^{\zeta}\cos(y)&-x_{1}e^{\zeta}\sin(y)&e^{\zeta+x}\cos(y)&-e^{\zeta+x}\sin(y)&g_{14}&k_{14}\\\ b_{14}&x_{1}e^{\zeta}\sin(y)&x_{1}e^{\zeta}\cos(y)&e^{\zeta+x}\sin(y)&e^{\zeta+x}\cos(y)&h_{14}&l_{14}\\\ 0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix}$
|  where $\begin{cases}\zeta=\lambda_{1}x+\lambda_{2}y\\\ c_{14}:\text{ an
expression containing parameters }x_{1},x,y\\\
a_{14},b_{14},d_{14},f_{14},m_{14},n_{14}:\text{ expressions containing
parameters }x_{2},x_{3},x,y\\\ g_{14},h_{14},k_{14},l_{14}:\text{ expressions
containing parameters }x_{1},x_{2},x_{3},x_{4},x_{5},x,y.\end{cases}$
$G_{15}$ |  $\begin{bmatrix}1&0&0&0&0&0&0\\\ 0&e^{y}\cos(x)&-e^{y}\sin(x)&0&0&c_{15}&m_{15}\\\ 0&e^{y}\sin(x)&e^{y}\cos(x)&0&0&d_{15}&n_{15}\\\ a_{15}&p_{15}&q_{15}&e^{y}\cos(x)&-e^{y}\sin(x)&f_{15}&k_{15}\\\ b_{15}&r_{15}&p_{15}&e^{y}\sin(x)&e^{y}\cos(x)&g_{15}&l_{15}\\\ 0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix}$
|  where $\begin{cases}p_{15}=(x_{1}\cos(x)-y\sin(x))e^{y}\\\
q_{15}=(-x_{1}\sin(x)-y\cos(x))e^{y}\\\ r_{15}=(x_{1}\sin(x)+y\cos(x))e^{y}\\\
a_{15},b_{15},c_{15},d_{15},m_{15},n_{15}:\text{ expressions containing
parameters }x_{2},x_{3},x,y\\\ f_{15},g_{15},,k_{15},l_{15}:\text{ expressions
containing parameters }x_{1},x_{2},x_{3},x_{4},x_{5},x,y.\end{cases}$
$G^{\lambda}_{16}$ |  $\begin{bmatrix}1&0&0&0&0&0&0\\\ 0&e^{y}\cos(x)&-e^{y}\sin(x)&0&0&c_{16}&m_{16}\\\ 0&e^{y}\sin(x)&e^{y}\cos(x)&0&0&d_{16}&n_{16}\\\ a_{16}&p_{16}&q_{16}&e^{y}\cos(x)&-e^{y}\sin(x)&f_{16}&k_{16}\\\ b_{16}&r_{16}&p_{16}&e^{y}\sin(x)&e^{y}\cos(x)&g_{16}&l_{16}\\\ 0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&1\end{bmatrix}$
|  where
$\begin{cases}p_{16}=\left(x_{1}\cos(x)-\dfrac{1}{2}x\sin(x)\right)e^{y}-\lambda
y\sin(x)e^{y}\\\
q_{16}=\left(-x_{1}\sin(x)-\dfrac{1}{2}x\cos(x)+\dfrac{1}{2}\sin(x)\right)e^{y}-\lambda
y\cos(x)e^{y}\\\
r_{16}=\left(x_{1}\sin(x)+\dfrac{1}{2}x\cos(x)+\dfrac{1}{2}\sin(x)\right)e^{y}+\lambda
y\cos(x)e^{y}\\\ a_{16},b_{16},c_{16},d_{16},m_{16},n_{16}:\text{ expressions
containing parameters }x_{2},x_{3},x,y\\\ f_{16},g_{16},k_{16},l_{16}:\text{
expressions containing parameters
}x_{1},x_{2},x_{3},x_{4},x_{5},x,y.\end{cases}$
Table 3: (continued)

∎

######  Remark 2.2 (Geometry of maximal-dimensional K-orbits).

Due to the picture of K-orbits of maximal dimension of considered Lie groups,
we have some geometric characteristics as follows.

  1. 1.

All K-orbits of $G\in\mathcal{A}$ are strictly homogeneous simplectic
submanifolds of $\mathcal{G}^{*}$ (see [8, §15.1]). Moreover, if $G$ is
exponential, i.e. $G\in\mathcal{A}_{1}$, all K-orbits of $G$ are homeomorphic
to Euclidean spaces (see [8, §15.3]). It can be easily verified by using the
picture of K-orbits in Theorem 1.

  2. 2.

For each $G\in\mathcal{A}_{1}$, there are exactly two types of maximal-
dimensional K-orbits.

    1. 2.1

For $G=G_{1}^{\lambda}$, Type 1 has exactly four K-orbits but Type 2 has an
infinite family of K-orbits. Namely,

       * i.

Each orbit of Type 1 is a quarter of hyperplane $\\{x_{4}^{*}=0\\}$ in
$\mathcal{G}^{*}$ which is obtained when “cutting” it by two hyperplanes
$\\{x_{2}^{*}=0\\}$ and $\\{x_{5}^{*}=0\\}$. In fact, K-orbits of Type 1 are
four connected components of
$\\{x_{4}^{*}=0\\}\setminus\left(\\{x_{2}^{*}=0\\}\cup\\{x_{5}^{*}=0\\}\right)$.

       * ii.

Each orbit of Type 2 is a part of hyperplane $\\{x_{4}^{*}=c\\}$ ($c$ is a
non-zero constant) which is obtained when “cutting” it by two hyperplanes
$\\{x_{4}^{*}=0\\}$ and $\\{x_{5}^{*}=0\\}$.

    2. 2.2

For $G=G_{12}^{\lambda}$, Type 1 has exactly four K-orbits but Type 2 has an
infinite family of K-orbits. Namely,

       * i.

Each orbit of Type 1 is a quarter of hyperplane $\\{x_{5}^{*}=0\\}$ in
$\mathcal{G}^{*}$ which is obtained when “cutting” it by two hyperplanes
$\\{x_{3}^{*}=0\\}$ and $\\{x_{4}^{*}=0\\}$. In fact, K-orbits of Type 1 are
four connected components of
$\\{x_{5}^{*}=0\\}\setminus\left(\\{x_{3}^{*}=0\\}\cup\\{x_{4}^{*}=0\\}\right)$.

       * ii.

Each orbit of Type 2 is a half of an algebraic hypersurface in
$\mathcal{G}^{*}$ which is obtained when “cutting” it by hyperplane
$\\{x_{5}^{*}=0\\}$.

    3. 2.3

For each $G\in\mathcal{A}_{1}\setminus\\{G_{1}^{\lambda},G_{12}^{\lambda}\\}$,
Type 1 has a finite family of K-orbits but Type 2 has an infinite family of
K-orbits. Namely,

       * i.

Type 1:

         * (i)

If $G\in\\{G_{2},G_{11}\\}$, each orbit of Type 1 is a quarter of hyperplane
$\\{x_{5}^{*}=0\\}$ in $\mathcal{G}^{*}$ which is obtained when “cutting” it
by two hyperplanes $\\{x_{3}^{*}=0\\}$ and $\\{x_{4}^{*}=0\\}$.

         * (ii)

If $G\in\left\\{G_{7},G^{\lambda}_{8}\right\\}$, each orbit of Type 1 is
either a haft of hyperplane $\\{x_{4}^{*}=0\\}$ in $\mathcal{G}^{*}$ which is
obtained when “cutting” it by hyperplane $\\{x_{5}^{*}=0\\}$ or a quarter of
hyperplane $\\{x_{5}^{*}=0\\}$ in $\mathcal{G}^{*}$ which is obtained when
“cutting” it by two hyperplanes $\\{x_{3}^{*}=0\\}$ and $\\{x_{4}^{*}=0\\}$.

         * (iii)

If
$G\in\left\\{G_{3},G_{4}^{\lambda_{1}\lambda_{2}},G_{5},G_{6}^{\lambda},G_{9},G_{10}^{\lambda}\right\\}$,
each orbit of Type 1 is either a quarter of hyperplane $\\{x_{4}^{*}=0\\}$ in
$\mathcal{G}^{*}$ which is obtained when “cutting” it by two hyperplanes
$\\{x_{2}^{*}=0\\}$ and $\\{x_{5}^{*}=0\\}$ or a quarter of hyperplane
$\\{x_{5}^{*}=0\\}$ in $\mathcal{G}^{*}$ which is obtained when “cutting” it
by two hyperplanes $\\{x_{3}^{*}=0\\}$ and $\\{x_{4}^{*}=0\\}$.

       * ii.

Each orbit of Type 2 is a part of a (transcendental or algebraic) hypersurface
in $\mathcal{G}^{*}$ which is obtained when “cutting” it by two hyperplanes
$\\{x_{4}^{*}=0\\}$ and $\\{x_{5}^{*}=0\\}$.

    4. 2.4

Now we consider in more detail one concrete group mentioned at the beginning
of this item (i.e. groups have two types of maximal-dimensional K-orbits),
such as $G=G_{4}^{\lambda_{1}\lambda_{2}}$. Denote by $\sim$ the equivalence
relation which is defined on $\mathcal{G}^{*}$ as follows: $F_{1}\sim
F_{2}\Leftrightarrow\Omega_{F_{1}}=\Omega_{F_{2}}$. Then each K-orbit of $G$
can be seen as an element of the space $\mathcal{G}^{*}/\sim$ with quotient
topology. Assume that

$F(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha,\beta),F_{0}(\alpha_{1},\alpha_{2},\alpha_{3},0,\alpha_{5},\alpha,\beta)\in\mathcal{G}^{*}$

with $\alpha_{i}\neq 0$ for $i=1,\ldots,5$. Note that $\Omega_{F}$ is of Type
2 while $\Omega_{F_{0}}$ is of Type 1. By letting $\alpha_{4}\rightarrow 0$,
we have

$F(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha,\beta)\rightarrow
F_{0}(\alpha_{1},\alpha_{2},\alpha_{3},0,\alpha_{5},\alpha,\beta).$

However, it is clear that the orbit $\Omega_{F}$ of $F$ does not converge to
$\Omega_{F_{0}}$ in $\mathcal{G}^{*}/\sim$. In other words, K-orbits of Type 1
seem to play a role as “singularity” and should be excluded from the family of
K-orbits in general position. Thus, the family of K-orbits in general position
are only K-orbits of Type 2. For all remaining groups
$G\in\mathcal{A}_{1}\setminus\\{G_{4}^{\lambda_{1}\lambda_{2}}\\}$, we also
have absolutely similar situations. In the following, we will formulate a
definition for general position K-orbits.

  3. 3.

For each group $G\in\mathcal{A}_{2}$ there is an infinite family of maximal-
dimensional K-orbits. All these orbits are of the same type. Namely, each of
them is always a certain (transcendental or algebraic) hypersurfaces in
$\mathcal{G}^{*}$. Of course, all K-orbits of $G\in\mathcal{A}_{2}$ are non-
singular, and they are all in general position.

Each maximal-dimensional K-orbit in the general position will be simply called
a _generic_ K-orbit. Namely, we have the following definition.

######  Definition 2.3.

Let $G\in\mathcal{A}$ ($=\mathcal{A}_{1}\sqcup\mathcal{A}_{2}$). A six-
dimensional K-orbit $\Omega$ of $G$ is called a _generic K-orbit_ if either
$G\in\mathcal{A}_{2}$, or $G\in\mathcal{A}_{1}$ and $\Omega$ is of Type 2.

######  Remark 2.4 (Geometric characteristics of generic K-orbits).

We will now examine more carefully the family $\mathcal{F}_{G}$ of all generic
K-orbits as in Definition 2.3. For convenience, we set

$\begin{array}[]{l
l}V_{G}:=\cup\left\\{\Omega\,|\,\Omega\in\mathcal{F}_{G}\right\\},&\text{for
}G\in\mathcal{A}.\end{array}$

By the picture of generic K-orbits of each group $G$, we can divide sixteen
families of Lie groups in $\mathcal{A}$ into three subfamilies such that each
subfamily has an almost similar picture of generic K-orbits. In the following,
we will describe the geometric characteristics of the generic K-orbits for
each subfamily.

  1. 1.

For $G\in\mathcal{A}_{1}\setminus\\{G^{\lambda}_{12}\\}$, we have

$V_{G}=V_{1}:=\left\\{(x^{*}_{1},\ldots,x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}\,\big{|}\,x^{*}_{4}x^{*}_{5}\neq
0\right\\}\subset\mathcal{G}^{*}\equiv\mathbb{R}^{7}.$

In fact,

$V_{1}\equiv\mathbb{R}^{3}\times\left(\mathbb{R}\setminus\\{0\\}\right)^{2}\times\mathbb{R}^{2}={V_{1}}_{++}\sqcup{V_{1}}_{-+}\sqcup{V_{1}}_{--}\sqcup{V_{1}}_{+-}$
(2.1)

where

${V_{1}}_{++}:=\mathbb{R}^{3}\times\mathbb{R}_{+}\times\mathbb{R}_{+}\times\mathbb{R}^{2},\quad{V_{1}}_{-+}:=\mathbb{R}^{3}\times\mathbb{R}_{-}\times\mathbb{R}_{+}\times\mathbb{R}^{2}$
(2.2)
${V_{1}}_{--}:=\mathbb{R}^{3}\times\mathbb{R}_{-}\times\mathbb{R}_{-}\times\mathbb{R}^{2},\quad{V_{1}}_{+-}:=\mathbb{R}^{3}\times\mathbb{R}_{+}\times\mathbb{R}_{-}\times\mathbb{R}^{2}$
(2.3)

in which $\mathbb{R}_{+}:=\\{x\in\mathbb{R}:x>0\\}$ and
$\mathbb{R}_{-}:=\\{x\in\mathbb{R}:x<0\\}$.

  2. 2.

If $G=G^{\lambda}_{12}$, we have

$V_{G}=V_{2}:=\left\\{(x^{*}_{1},\dots,x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}\,\big{|}\,x^{*}_{5}\neq
0\right\\}\subset\mathcal{G}^{*}\equiv\mathbb{R}^{7}.$

In fact,

$V_{2}\equiv\mathbb{R}^{4}\times\left(\mathbb{R}\setminus\\{0\\}\right)\times\mathbb{R}^{2}={V_{2}}_{+}\sqcup{V_{2}}_{-}$
(2.4)

where

${V_{2}}_{+}:=\mathbb{R}^{4}\times\mathbb{R}_{+}\times\mathbb{R}^{2},\quad{V_{2}}_{-}:=\mathbb{R}^{4}\times\mathbb{R}_{-}\times\mathbb{R}^{2}.$
(2.5)

  3. 3.

For $G\in\mathcal{A}_{2}$, we have

$V_{G}=V_{3}:=\left\\{(x^{*}_{1},\dots,x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}\,\big{|}\,{x_{4}^{*}}^{2}+{x_{5}^{*}}^{2}\neq
0\right\\}\subset\mathcal{G}^{*}\equiv\mathbb{R}^{7}.$

In fact

$V_{3}\equiv\mathbb{R}^{3}\times\left(\mathbb{R}^{2}\setminus\\{(0,0)\\}\right)\times\mathbb{R}^{2}\subset\mathcal{G}^{*}\equiv\mathbb{R}^{7}.$
(2.6)

Obviously, all $V_{G}$ above are open submanifolds of
$\mathcal{G}^{*}\equiv\mathbb{R}^{7}$ (with natural differential structure),
and each K-orbit $\Omega$ from $\mathcal{F}_{G}$ is a six-dimensional
submanifold of $V_{G}$. In the next section, we will prove that
$\mathcal{F}_{G}$ forms a measurable foliation on the open submanifold
$V_{G}$.

Table 4: The picture of maximal-dimensional K-orbits Conditions for $\dim\Omega_{F}=6$ |  $G$ |  $\Omega_{F}=\left\\{F_{U}(x_{1}^{*},\dots,x_{5}^{*},x^{*},y^{*})\right\\}$ |  Geometric forms |  Types
---|---|---|---|---
$\begin{array}[]{l}\alpha_{5}\neq 0\\\ \alpha_{2}^{2}+\alpha_{4}^{2}\neq 0\\\ \alpha_{1},\alpha_{3},\alpha,\beta\in\mathbb{R}\end{array}$ |  $\alpha_{2}\alpha_{5}\neq 0=\alpha_{4}$ |  $G^{\lambda}_{1}$ |  $x_{4}^{*}=0$, $\alpha_{2}x^{*}_{2}>0$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x_{3}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of hyperplane $\\{x^{*}_{4}=0\\}$ |  singularity
$\alpha_{4}\alpha_{5}\neq 0$ |  $x^{*}_{4}=\alpha_{4}$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x_{2}^{*},x_{3}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of hyperplane $\\{x^{*}_{4}=\alpha_{4}\\}$ |  generic
$\begin{array}[]{l}\alpha_{4}\neq 0,\\\ \alpha_{3}^{2}+\alpha_{5}^{2}\neq 0,\\\ \alpha_{1},\alpha_{2},\alpha,\beta\in\mathbb{R}\end{array}$ |  $\alpha_{3}\alpha_{4}\neq 0=\alpha_{5}$ |  $G_{2}$, $G_{11}$ |  $x_{5}^{*}=0$, $\alpha_{3}x^{*}_{3}>0$, $\alpha_{4}x^{*}_{4}>0$, $x_{1}^{*},x_{2}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of hyperplane $\\{x^{*}_{5}=0\\}$ |  singularity
$\alpha_{4}\alpha_{5}\neq 0$ |  $G_{2}$ |  $x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}=\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}$, $\alpha_{4}x^{*}_{4}>0,\alpha_{5}x^{*}_{5}>0,x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of an algebraic hypersurface |  generic
$G_{11}$ |  $\left(\frac{x^{*}_{2}}{x_{5}}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*2}_{5}}\right)e^{\frac{x^{*}_{4}}{x^{*}_{5}}}=\left(\frac{\alpha_{2}}{\alpha_{5}}-\frac{\alpha_{3}\alpha_{4}}{\alpha^{2}_{5}}\right)e^{\frac{\alpha_{4}}{\alpha_{5}}}$, $\alpha_{4}x^{*}_{4}>0$, $\alpha_{5}x^{*}_{5}>0,x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of a transcendental hypersurface |  generic
$\begin{array}[]{l}\alpha_{4}^{2}+\alpha_{5}^{2}\neq 0,\\\ \text{except for}\\\ \alpha_{3}=\alpha_{5}=0\neq\alpha_{4},\\\ \alpha_{1},\alpha_{2},\alpha,\beta\in\mathbb{R}\end{array}$ |  $\alpha_{4}=0\neq\alpha_{5}$ |  $G_{7}$, $G_{8}$ |  $x^{*}_{4}=0$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x_{2}^{*},x_{3}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of hyperplane $\\{x^{*}_{4}=0\\}$ |  singularity
$\alpha_{3}\alpha_{4}\neq 0=\alpha_{5}$ |  $x_{5}^{*}=0$, $\alpha_{3}x^{*}_{3}>0$, $\alpha_{4}x^{*}_{4}>0$, $x_{1}^{*},x_{2}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of hyperplane $\\{x^{*}_{5}=0\\}$ |  singularity
$\alpha_{4}\alpha_{5}\neq 0$ |  $G_{7}$ |  $\frac{x^{*}_{2}}{x^{*}_{5}}-\frac{x^{*}_{3}x^{*}_{4}}{{x^{*}_{5}}^{2}}-\ln|x^{*}_{4}|+\ln|x^{*}_{5}|=\frac{\alpha_{2}}{\alpha_{5}}-\frac{\alpha_{3}\alpha_{4}}{\alpha^{2}_{5}}-\ln|\alpha_{4}|+\ln|\alpha_{5}|$, $\alpha_{4}x^{*}_{4}>0$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of a transcendental hypersurface |  generic
$G_{8}^{\lambda}$ |  $\frac{x^{*}_{2}}{x^{*}_{5}}-\frac{x^{*}_{3}x^{*}_{4}}{{x^{*}_{5}}^{2}}-(2+\lambda)\ln|x^{*}_{5}|+(1+\lambda)\ln|x^{*}_{4}|=\frac{\alpha_{2}}{\alpha_{5}}-\frac{\alpha_{3}\alpha_{4}}{\alpha^{2}_{5}}-(2+\lambda)\ln|\alpha_{5}|+(1+\lambda)\ln|\alpha_{4}|$, $\alpha_{4}x^{*}_{4}>0$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of a transcendental hypersurface |  generic
$\begin{array}[]{l}\alpha_{4}=0\neq\alpha_{2}\alpha_{5},\text{or }\\\ \alpha_{4}\neq 0\neq\alpha_{3}^{2}+\alpha_{5}^{2},\\\ \alpha_{1},\alpha,\beta\in\mathbb{R}\end{array}$ |  $\alpha_{4}=0\neq\alpha_{2}\alpha_{5}$ |  $G_{3}$, $G_{4}^{\lambda_{1}\lambda_{2}}$, $G_{5}$, $G_{6}^{\lambda}$, $G_{9}$, $G_{10}^{\lambda}$ |  $x_{4}^{*}=0$, $\alpha_{2}x^{*}_{2}>0$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x_{3}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of hyperplane $\\{x^{*}_{4}=0\\}$ |  singularity
$\alpha_{5}=0\neq\alpha_{3}\alpha_{4}$ |  |  $x_{5}^{*}=0$, $\alpha_{3}x^{*}_{3}>0$, $\alpha_{4}x^{*}_{4}>0$, $x_{1}^{*},x_{2}^{*},x^{*},y^{*}\in\mathbb{R}$ |  part of hyperplane $\\{x^{*}_{5}=0\\}$ |  singularity
|  $G_{3}$ |  $\frac{x^{*}_{2}}{x^{*}_{4}}-\frac{x^{*}_{3}}{x^{*}_{5}}=\frac{\alpha_{2}}{\alpha_{4}}-\frac{\alpha_{3}}{\alpha_{5}}$, $\alpha_{4}x^{*}_{4}>0$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of an algebraic hypersurface |  generic
|  $G_{4}^{\lambda_{1}\lambda_{2}}$ |  $\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}=\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{{\alpha_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{\alpha_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}$, $\alpha_{4}x^{*}_{4}>0$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of an algebraic hypersurface |  generic
$\alpha_{4}\alpha_{5}\neq 0$ |  $G_{5},G_{6}^{\lambda}$ |  $\left(\frac{x^{*}_{2}}{x^{*}_{4}}-\frac{x^{*}_{3}}{x^{*}_{5}}\right)\sqrt{|x^{*}_{4}|}=\left(\frac{\alpha_{2}}{\alpha_{4}}-\frac{\alpha_{3}}{\alpha_{5}}\right)\sqrt{|\alpha_{4}|}$, $\alpha_{4}x^{*}_{4}>0$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of an algebraic hypersurface |  generic
|  $G_{9}$ |  $\frac{x^{*}_{2}}{x^{*}_{4}}-\frac{x^{*}_{3}}{x^{*}_{5}}+\ln|x^{*}_{4}|=\frac{\alpha_{2}}{\alpha_{4}}-\dfrac{\alpha_{3}}{\alpha_{5}}+\ln|\alpha_{4}|$, $\alpha_{4}x^{*}_{4}>0$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of a transcendental hypersurface |  generic
|  $G_{10}^{\lambda}$ |  $\frac{x^{*}_{2}}{x^{*}_{4}}-\frac{x^{*}_{3}}{x^{*}_{5}}-\lambda\ln|x^{*}_{4}|+\ln|x^{*}_{5}|=\frac{\alpha_{2}}{\alpha_{4}}-\frac{\alpha_{3}}{\alpha_{5}}-\lambda\ln|\alpha_{4}|+\ln|\alpha_{5}|$, $\alpha_{4}x^{*}_{4}>0$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of a transcendental hypersurface |  generic
$\begin{array}[]{l}\alpha_{3}\alpha_{4}\neq 0=\alpha_{5},\\\ \text{or }\alpha_{5}\neq 0,\\\ \alpha_{1},\alpha_{2},\alpha,\beta\in\mathbb{R}\end{array}$ |  $\alpha_{3}\alpha_{4}\neq 0=\alpha_{5}$ |  $G_{12}^{\lambda}$ |  $x^{*}_{5}=0$, $\alpha_{3}x^{*}_{3}>0$, $\alpha_{4}x^{*}_{4}>0$, $x_{1}^{*},x_{2}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a part of the hyperplane $\\{x^{*}_{5}=0\\}$ |  singularity
$\alpha_{5}\neq 0$ |  $\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{1}{{x^{*}_{5}}^{\frac{\lambda}{1+\lambda}}e^{\frac{x^{*}_{4}}{(1+\lambda)x^{*}_{5}}}}=\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{1}{\alpha_{5}^{\frac{\lambda}{1+\lambda}}e^{\frac{\alpha_{4}}{(1+\lambda)\alpha_{5}}}}$, $\alpha_{5}x^{*}_{5}>0$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$ |  a half of a transcendental hypersurface |  generic
$G^{\lambda}_{13}$ |  $\frac{x^{*}_{2}x^{*}_{5}-x^{*}_{3}x^{*}_{4}}{{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}}e^{\lambda\arctan\frac{x^{*}_{4}}{x^{*}_{5}}}=\frac{\alpha_{2}\alpha_{5}-\alpha_{3}\alpha_{4}}{{\alpha_{4}}^{2}+{\alpha_{5}}^{2}}e^{\lambda\arctan\frac{\alpha_{4}}{\alpha_{5}}}$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$, $x_{4}^{*}x_{5}^{*}\neq 0$ |  |
$\frac{x^{*}_{2}}{x_{5}^{*}}=\frac{\alpha_{2}\alpha_{5}-\alpha_{3}\alpha_{4}}{{\alpha_{4}}^{2}+{\alpha_{5}}^{2}}e^{\lambda\arctan\frac{\alpha_{4}}{\alpha_{5}}}$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$, $x_{4}^{*}=0\neq x_{5}^{*}$ |  |
$\begin{array}[]{l}\alpha_{4}^{2}+\alpha_{5}^{2}\neq 0,\\\ \alpha_{1},\alpha_{2},\alpha_{3},\alpha,\beta\in\mathbb{R}\end{array}$ |  $G_{14}^{\lambda_{1}\lambda_{2}}$ |  $\frac{x^{*}_{3}x^{*}_{4}-x^{*}_{2}x^{*}_{5}}{\left({x_{4}^{*}}^{2}+{x_{5}^{*}}^{2}\right)^{\frac{2\lambda_{1}+1}{2(1+\lambda_{1})}}}e^{\frac{\lambda_{2}}{1+\lambda_{1}}\arctan\frac{x^{*}_{5}}{x^{*}_{4}}}=\frac{\alpha_{3}\alpha_{4}-\alpha_{2}\alpha_{5}}{\left(\alpha_{4}^{2}+\alpha_{5}^{2}\right)^{\frac{2\lambda_{1}+1}{2(1+\lambda_{1})}}}e^{\frac{\lambda_{2}}{1+\lambda_{1}}\arctan\frac{\alpha_{5}}{\alpha_{4}}}$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$, $x_{4}^{*}x_{5}^{*}\neq 0$ |  |
|  |  $\frac{x^{*}_{3}}{{x^{*}}_{4}^{\frac{\lambda_{1}}{1+\lambda_{1}}}}=\frac{\alpha_{3}\alpha_{4}-\alpha_{2}\alpha_{5}}{\left(\alpha_{4}^{2}+\alpha_{5}^{2}\right)^{\frac{2\lambda_{1}+1}{2(1+\lambda_{1})}}}e^{\frac{\lambda_{2}}{1+\lambda_{1}}\arctan\frac{\alpha_{5}}{\alpha_{4}}}$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$, $x_{4}^{*}\neq 0=x_{5}^{*}$ |  transcendental hypersurface |  generic
|  $G_{15}$ |  $\frac{x^{*}_{3}x^{*}_{4}-x^{*}_{2}x^{*}_{5}}{{x_{4}^{*}}^{2}+{x_{5}^{*}}^{2}}+\frac{\ln({x_{4}^{*}}^{2}+{x_{5}^{*}}^{2})}{2}=\frac{\alpha_{3}\alpha_{4}-\alpha_{2}\alpha_{5}}{\alpha_{4}^{2}+\alpha_{5}^{2}}+\frac{\ln(\alpha_{4}^{2}+\alpha_{5}^{2})}{2}$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$, $x_{4}^{*}x_{5}^{*}\neq 0$ |  |
|  $-\frac{x^{*}_{2}}{x^{*}_{5}}+\ln|x^{*}_{5}|=\frac{\alpha_{3}\alpha_{4}-\alpha_{2}\alpha_{5}}{\alpha_{4}^{2}+\alpha_{5}^{2}}+\frac{\ln(\alpha_{4}^{2}+\alpha_{5}^{2})}{2}$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$, $x_{4}^{*}=0\neq x_{5}^{*}$ |  |
|  $G_{16}^{\lambda}$ |  $\frac{x^{*}_{3}x^{*}_{4}-x^{*}_{2}x^{*}_{5}}{{x_{4}^{*}}^{2}+{x_{5}^{*}}^{2}}+\frac{x^{*}_{5}x^{*}_{4}}{2\left({x_{4}^{*}}^{2}+{x_{5}^{*}}^{2}\right)}+\frac{\lambda\ln\left({x^{*}}^{2}_{4}+{x^{*}}^{2}_{5}\right)}{2}-\frac{\arctan\frac{x^{*}_{5}}{x^{*}_{4}}}{2}=\frac{\alpha_{3}\alpha_{4}-\alpha_{2}\alpha_{5}}{\alpha^{2}_{4}+\alpha_{5}^{2}}+\frac{\alpha_{5}\alpha_{4}}{2(\alpha^{2}_{4}+\alpha_{5}^{2})}+\frac{\lambda\ln\left({\alpha}^{2}_{4}+{\alpha}^{2}_{5}\right)-\arctan\frac{\alpha_{5}}{\alpha_{4}}}{2},$ $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$, $x_{4}^{*}x_{5}^{*}\neq 0$ |  |
|  $\frac{x^{*}_{3}}{x^{*}_{4}}+\lambda\ln|x^{*}_{4}|=\frac{\alpha_{3}\alpha_{4}-\alpha_{2}\alpha_{5}}{\alpha^{2}_{4}+\alpha_{5}^{2}}+\frac{\alpha_{5}\alpha_{4}}{2(\alpha^{2}_{4}+\alpha_{5}^{2})}+\frac{\lambda\ln\left({\alpha}^{2}_{4}+{\alpha}^{2}_{5}\right)}{2}-\frac{\arctan\frac{\alpha_{5}}{\alpha_{4}}}{2}$, $x_{1}^{*},x^{*},y^{*}\in\mathbb{R}$, $x_{4}^{*}\neq 0=x_{5}^{*}$ |  |

##  3 Foliations formed by generic K-orbits of considered Lie groups

This section establishes the remaining results of the paper on foliations
formed by generic K-orbits of each Lie group $G\in\mathcal{A}$. Recall that
all $V_{G}\in\\{V_{1},V_{2},V_{3}\\}$ determining by (2.1), (2.4), (2.6) are
open submanifolds of $\mathcal{G}^{*}\equiv\mathbb{R}^{7}$, and
$\mathcal{F}_{G}$ is the family of all generic K-orbits of $G$ (see Definition
2.3).

######  Theorem 2.

For each group $G\in\mathcal{A}$, the family $\mathcal{F}_{G}$ of all generic
K-orbits of $G$ forms a measurable foliation on the open manifold $V_{G}$ in
the sense of Connes [1], and it is called the foliation associated with $G$.

###### Proof.

The proof is analogous to the case of MD-groups in [18, 22, 23, 24] and Lie
groups in [17, Theorem 21]. It consists of two steps as follows

  * 1.

Step 1: For any $G\in\mathcal{A}$, construct a suitable differential system
$S_{G}$ of rank six on the manifold $V_{G}$ such that each K-orbit $\Omega$
from $\mathcal{F}_{G}$ is a maximal connected integral submanifold
corresponding to $S_{G}$.

  * 2.

Step 2: Afterwards, show that the Lebesgue measure is invariant for some
smooth polyvector field $\mathfrak{X}$ of degree six such that it generates
$S_{G}$.

As emphasized in Remark 2.4, sixteen families of Lie groups in $\mathcal{A}$
can be divided into three subfamilies as follows

$\mathcal{A}=\left\\{G^{\lambda}_{1},G_{2},G_{3},G_{4}^{\lambda_{1}\lambda_{2}},G_{5},G_{6}^{\lambda},G_{7},G_{8}^{\lambda},G_{9},G^{\lambda}_{10},G_{11}\right\\}\cup\left\\{G_{12}^{\lambda}\right\\}\cup\left\\{G^{\lambda}_{13},G_{14}^{\lambda_{1}\lambda_{2}},G_{15},G_{16}^{\lambda}\right\\},$

where groups in the same subfamily have almost the same picture of generic
K-orbits. To prove Theorem 2, we thus choose in each of these subfamilies a
representative group and calculate carefully for that selected group.

Note that the assertion of Theorem 2 for cases $G_{2}$, $G_{3}$, $G_{9}$ and
$G^{\lambda}_{10}$ had been proved in [17]. In the following, we choose three
groups $G_{4}^{\lambda_{1}\lambda_{2}}$, $G_{12}^{\lambda}$ and
$G_{13}^{\lambda}$ and prove Theorem 2 explicitly for this cases.

Let $G=G_{4}^{\lambda_{1}\lambda_{2}}$. Proof of Step 1. For any
$F(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},,\alpha_{5},\alpha,\beta)\in\mathcal{G}^{*}$,
by Theorem 1, the generic K-orbit $\Omega_{F}$ belongs to $\mathcal{F}_{G}$ if
and only if $\alpha_{4}\alpha_{5}\neq 0$. Furthermore, if we denote by $v$ the
element
$(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}$
then

$\displaystyle\Omega_{F}=\Bigg{\\{}v\in\mathcal{G}^{*}\,\bigg{|}\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}=\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{{\alpha_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{\alpha_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}},\alpha_{4}x^{*}_{4}>0,\,\alpha_{5}x^{*}_{5}>0\Bigg{\\}}.$

Now we consider on $V_{G}$ the following differential system $S_{G}$:

$\begin{cases}\mathfrak{X}_{1}:=\frac{\partial}{\partial x^{*}_{1}}\\\
\mathfrak{X}_{2}:=\lambda_{1}x^{*}_{3}\frac{\partial}{\partial
x^{*}_{3}}+x^{*}_{4}\frac{\partial}{\partial
x^{*}_{4}}+(\lambda_{1}+1)x^{*}_{5}\frac{\partial}{\partial x^{*}_{5}}\\\
\mathfrak{X}_{3}:=x^{*}_{2}\frac{\partial}{\partial
x^{*}_{2}}+\lambda_{2}x^{*}_{3}\frac{\partial}{\partial
x^{*}_{3}}+x^{*}_{4}\frac{\partial}{\partial
x^{*}_{4}}+\lambda_{2}x^{*}_{5}\frac{\partial}{\partial x^{*}_{5}}\\\
\mathfrak{X}_{4}:=x^{*}_{4}\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{5}\frac{\partial}{\partial x^{*}_{3}}\\\
\mathfrak{X}_{5}:=\frac{\partial}{\partial x^{*}}\\\
\mathfrak{X}_{6}:=\frac{\partial}{\partial y^{*}}.\end{cases}$

Obviously, $\operatorname{rank}(S_{G})=6$ and all
$\mathfrak{X}_{i}\,(i=1,\ldots,6)$ are smooth over $V_{G}$. We claim that
$S_{G}$ generates $\mathcal{F}_{G}$, i.e. each K-orbit $\Omega$ from
$\mathcal{F}_{G}$ is a maximal connected integral submanifold of $S_{G}$.

If fact, we first consider
$\mathfrak{X}_{1},\mathfrak{X}_{5},\mathfrak{X}_{6}$. Clearly, their flows
(i.e. one-parameter subgroups) are determined as follows

$\begin{array}[]{l
l}\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}:&F\mapsto\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}(F):=(x^{*}_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha,\beta),\\\
\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}:&F\mapsto\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}(F):=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},x^{*},\beta),\\\
\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}:&F\mapsto\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}(F):=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha,y^{*}).\end{array}$

Next, we consider $\mathfrak{X}_{2}$. For some positive
$\epsilon\in\mathbb{R}$, assume that

$\varphi:t\mapsto\varphi(t)=\left(x^{*}_{1}(t),x^{*}_{2}(t),x^{*}_{3}(t),x^{*}_{4}(t),x^{*}_{5}(t),x^{*}(t),y^{*}(t)\right),\quad
t\in(-\epsilon,\epsilon)$

is an integral curve of $\mathfrak{X}_{2}$ passing $F=\varphi(0)$. Then, we
must have $\varphi^{\prime}(t)={\mathfrak{X}_{2}}_{\varphi(t)}$ and it is
equivalent to

$\begin{array}[]{l}\sum\limits_{i=1}^{5}x^{*^{\prime}}_{i}(t)\frac{\partial}{\partial
x^{*}_{i}}+x^{*^{\prime}}(t)\frac{\partial}{\partial
x^{*}}+y^{*^{\prime}}(t)\frac{\partial}{\partial
y^{*}}=\lambda_{1}x^{*}_{3}(t)\frac{\partial}{\partial
x^{*}_{3}}+x^{*}_{4}(t)\frac{\partial}{\partial
x^{*}_{4}}+(1+\lambda_{1})x^{*}_{5}(t)\frac{\partial}{\partial
x^{*}_{5}}\cdot\end{array}$

By identifying coefficients on two sides, we obtain

$\begin{cases}x^{*^{\prime}}_{1}(t)=x^{*^{\prime}}_{2}(t)=x^{*^{\prime}}(t)=y^{*^{\prime}}(t)=0\\\
x^{*^{\prime}}_{3}(t)=\lambda_{1}x^{*}_{3}(t)\\\
x^{*^{\prime}}_{4}(t)=x^{*}_{4}(t)\\\
x^{*^{\prime}}_{5}(t)=(1+\lambda_{1})x^{*}_{5}(t).\end{cases}$ (3.1)

Since $F=\varphi(0)$, system (3) gives us

$x^{*}_{1}=\alpha_{1},\,x^{*}_{2}=\alpha_{2},\,x^{*}_{3}=\alpha_{3}e^{\lambda_{1}t},\,x^{*}_{4}=\alpha_{4}e^{t},\,x^{*}_{5}=\alpha_{5}e^{(1+\lambda_{1})t},\,x^{*}=\alpha,\,y^{*}=\beta.$

Therefore, the flow of $\mathfrak{X}_{2}$ is as follows

$\theta^{\mathfrak{X}_{2}}_{x}:F\mapsto\theta^{\mathfrak{X}_{2}}_{x}(F):=(\alpha_{1},\alpha_{2},\alpha_{3}e^{\lambda_{1}x},\alpha_{4}e^{x},\alpha_{5}e^{(1+\lambda_{1})x},\alpha,\beta).$

Similarly, the flows of $\mathfrak{X}_{3}$ is as follows

$\theta^{\mathfrak{X}_{3}}_{y}:F\mapsto\theta^{\mathfrak{X}_{3}}_{y}(F):=(\alpha_{1},\alpha_{2}e^{y},\alpha_{3}e^{\lambda_{2}y},\alpha_{4}e^{y},\alpha_{5}e^{\lambda_{2}y},\alpha,\beta).$

Now, we consider $\mathfrak{X}_{4}$. Assume that

$\varphi:t\mapsto\varphi(t)=(x^{*}_{1}(t),x^{*}_{2}(t),x^{*}_{3}(t),x^{*}_{4}(t),x^{*}_{5}(t),x^{*}(t),y^{*}(t))$

be an integral curve of $\mathfrak{X}_{4}$ passing $F=\varphi(0)$, where
$t\in(-\epsilon,\epsilon)\subset\mathbb{R}$ for some positive
$\epsilon\in\mathbb{R}$. Then we have
$\varphi^{\prime}(t)={\mathfrak{X}_{4}}_{\varphi(t)}$ which is equivalent to

$\displaystyle\sum_{i=1}^{5}x^{*^{\prime}}_{i}(t)\frac{\partial}{\partial
x^{*}_{i}}+x^{*^{\prime}}(t)\frac{\partial}{\partial
x^{*}}+y^{*^{\prime}}(t)\frac{\partial}{\partial
y^{*}}=x^{*}_{4}(t)\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{5}(t)\frac{\partial}{\partial x^{*}_{3}}$
$\displaystyle\Leftrightarrow$
$\displaystyle\begin{cases}x^{*^{\prime}}_{1}(t)=x^{*^{\prime}}_{4}(t)=x^{*^{\prime}}_{5}(t)=x^{*^{\prime}}(t)=y^{*^{\prime}}(t)=0\\\
x^{*^{\prime}}_{2}(t)=x^{*}_{4}(t)\\\
x^{*^{\prime}}_{3}(t)=x^{*}_{5}(t).\end{cases}$ (3.2)

Since $F=\varphi(0)$, equation (3) gives us

$x^{*}_{1}=\alpha_{1},\,x^{*}_{2}=\alpha_{2}+\alpha_{4}t,\,x^{*}_{3}=\alpha_{3}+\alpha_{5}t,\,x^{*}_{4}=\alpha_{4},\,x^{*}_{5}=\alpha_{5},\,x^{*}=\alpha,\;y^{*}=\beta.$

Therefore, the flow of $\mathfrak{X}_{4}$ is as follows

$\theta^{\mathfrak{X}_{4}}_{x_{1}}:F\mapsto\theta^{\mathfrak{X}_{2}}_{x_{1}}(F):=(\alpha_{1},\alpha_{2}+\alpha_{4}x_{1},\alpha_{3}+\alpha_{5}x_{1},\alpha_{4},\alpha_{5},\alpha,\beta).$

Finally, we set
$\theta=\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}\circ\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}\circ\theta^{\mathfrak{X}_{4}}_{x_{1}}\circ\theta^{\mathfrak{X}_{3}}_{y}\circ\theta^{\mathfrak{X}_{2}}_{x}\circ\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}$.
Then

$\displaystyle\theta(F)$
$\displaystyle=\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}\circ\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}\circ\theta^{\mathfrak{X}_{4}}_{x_{1}}\circ\theta^{\mathfrak{X}_{3}}_{y}\circ\theta^{\mathfrak{X}_{2}}_{x}\circ\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}(F)$
$\displaystyle=(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})$

where $x^{*}_{1},x^{*},y^{*}\in\mathbb{R}$ and

$\begin{array}[]{l
l}x^{*}_{2}=\alpha_{2}e^{y}+\alpha_{4}x_{1}e^{y},&x^{*}_{3}=\alpha_{3}e^{\lambda_{1}x+\lambda_{2}y}+\alpha_{5}x_{1}e^{\lambda_{1}x+\lambda_{2}y},\\\
x^{*}_{4}=\alpha_{4}e^{x+y},&x^{*}_{5}=\alpha_{5}e^{\lambda_{1}x+x+\lambda_{2}y}.\end{array}$

By direct calculations, we get

$\begin{array}[]{l}x^{*}_{2}=\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}+\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{{\alpha_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{\alpha_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}\frac{{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}\Leftrightarrow\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}=\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{{\alpha_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{\alpha_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}\cdot\end{array}$

Hence,
$\left\\{\theta(F)\,|\,x^{*}_{1},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\in\mathbb{R},\alpha_{4}x^{*}_{4}>0,\alpha_{5}x^{*}_{5}>0\right\\}\equiv\Omega_{F}$,
i.e. $\Omega_{F}$ is a maximal connected integral submanifold corresponding to
$S_{G}$. In other words, $S_{G}$ generates $\mathcal{F}_{G}$ and
$(V_{G},\mathcal{F}_{G})$ is a six-dimensional foliation for
$G=G_{4}^{\lambda_{1}\lambda_{2}}$, where $V_{G}=V_{1}$ as in (2.1).

Proof of Step 2. We have to show that the foliation $(V_{G},\mathcal{F}_{G})$
is measurable in the sense of Connes. As mentioned in [17, Subsection 2.2], to
prove that $(V_{G},\mathcal{F}_{G})$ is measurable, we only need to choose
some suitable pair ($\mathfrak{X},\mu$) on $V_{G}$, where $\mathfrak{X}$ is
some smooth 6-vector field defined on $V_{G}$, $\mu$ is some measure on
$V_{G}$ such that $\mathfrak{X}$ generates $S_{G}$ and $\mu$ is
$\mathfrak{X}$-invariant. Namely, we choose $\mu$ to be exactly the Lebesgue
measure on $V_{G}$ and set
$\mathfrak{X}:=\mathfrak{X}_{1}\wedge\mathfrak{X}_{2}\wedge\mathfrak{X}_{3}\wedge\mathfrak{X}_{4}\wedge\mathfrak{X}_{5}\wedge\mathfrak{X}_{6}$.
Clearly, $\mathfrak{X}$ is smooth, non-zero everywhere on $V_{G}$ and it is
exactly a polyvector field of degree six. Moreover, $\mathfrak{X}$ generates
$S_{G}$. In other words, if we choose a suitable orientation on
$(V_{G},\mathcal{F}_{G})$ then $\mathfrak{X}\in
C^{\infty}\left(\Lambda^{6}(\mathcal{F})\right)^{+}$. Besides, the invariance
of the Lebesgue measure $\mu$ with respect to $\mathfrak{X}$ is equivalent to
the invariance of $\mu$ for the K-representation that is restricted to the
foliated submanifold $V_{G}$ in $\mathcal{G}^{*}$. For any
$U(x_{1},x_{2},x_{3},x_{4},x_{5},x,y)\in\mathcal{G}$, direct computations show
that the Jacobi determinant $J_{U}$ of differential map
$K\left(\exp_{G}(U)\right)$ is a constant which depends only on $U$ but does
not depend on the coordinates of any point which moves in each generic K-orbit
$\Omega\in\mathcal{F}_{G}$. This means that the Lebegue measure $\mu$ is
$\mathfrak{X}$-invariant. This completes the proof for the case
$G=G_{4}^{\lambda_{1}\lambda_{2}}$.

Next, we consider the case $G=G_{12}^{\lambda}$. Proof of Step 1. For any
$F(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},,\alpha_{5},\alpha,\beta)\in\mathcal{G}^{*}$,
by Theorem 1, the generic K-orbits $\Omega_{F}$ belongs to $\mathcal{F}_{G}$
if and only if $\alpha_{5}\neq 0$. Furthermore, if we denote by $v$ the
element
$(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}$
then

$\begin{array}[]{r}\Omega_{F}=\Bigg{\\{}v\in\mathcal{G}^{*}\,\bigg{|}\,\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{1}{{x^{*}_{5}}^{\frac{\lambda}{1+\lambda}}e^{\frac{x^{*}_{4}}{(1+\lambda)x^{*}_{5}}}}=\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{1}{\alpha_{5}^{\frac{\lambda}{1+\lambda}}e^{\frac{\alpha_{4}}{(1+\lambda)\alpha_{5}}}},\;\alpha_{5}x^{*}_{5}>0\Bigg{\\}}.\end{array}$

On the open submanifold $V_{G}$ we consider the following differential system
$S_{G}$:

$\begin{cases}\mathfrak{X}_{1}:=\frac{\partial}{\partial x^{*}_{1}}\\\
\mathfrak{X}_{2}:=\lambda x^{*}_{2}\frac{\partial}{\partial x^{*}_{2}}+\lambda
x^{*}_{3}\frac{\partial}{\partial
x^{*}_{3}}+(\lambda+1)x^{*}_{4}\frac{\partial}{\partial
x^{*}_{4}}+(\lambda+1)x^{*}_{5}\frac{\partial}{\partial x^{*}_{5}}\\\
\mathfrak{X}_{3}:=(x^{*}_{2}+x^{*}_{3})\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{3}\frac{\partial}{\partial
x^{*}_{3}}+(x^{*}_{4}+x^{*}_{5})\frac{\partial}{\partial
x^{*}_{4}}+x^{*}_{5}\frac{\partial}{\partial x^{*}_{5}}\\\
\mathfrak{X}_{4}:=x^{*}_{4}\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{5}\frac{\partial}{\partial x^{*}_{3}}\\\
\mathfrak{X}_{5}:=\frac{\partial}{\partial x^{*}}\\\
\mathfrak{X}_{6}:=\frac{\partial}{\partial y^{*}}\cdot\end{cases}$

Obviously, $\operatorname{rank}(S_{G})=6$ and all
$\mathfrak{X}_{i}\,(i=1,\ldots,6)$ are smooth over $V_{G}$. Now, we will show
that $S_{G}$ generates $\mathcal{F}_{G}$.

First, we consider $\mathfrak{X}_{1},\mathfrak{X}_{5},\mathfrak{X}_{6}$. Their
flows (i.e. one-parameter subgroups) are determined as follows

$\begin{array}[]{ll}\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}:&F\mapsto\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}(F):=(x^{*}_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha,\beta)\\\
\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}:&F\mapsto\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}(F):=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},x^{*},\beta)\\\
\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}:&F\mapsto\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}(F):=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha,y^{*}).\end{array}$

Next, we consider $\mathfrak{X}_{2}:=\lambda x^{*}_{2}\frac{\partial}{\partial
x^{*}_{2}}+\lambda x^{*}_{3}\frac{\partial}{\partial
x^{*}_{3}}+(\lambda+1)x^{*}_{4}\frac{\partial}{\partial
x^{*}_{4}}+(\lambda+1)x^{*}_{5}\frac{\partial}{\partial x^{*}_{5}}$. For some
positive $\epsilon\in{\mathbb{R}}$, assume that

$\varphi:t\mapsto\varphi(t)=\bigl{(}x^{*}_{1}(t),x^{*}_{2}(t),x^{*}_{3}(t),x^{*}_{4}(t),x^{*}_{5}(t),x^{*}(t),y^{*}(t)\bigr{)};\;t\in(-\epsilon,\epsilon)$

is an integral curve of $\mathfrak{X}_{2}$ passing $F=\varphi(0)$. Then,
$\varphi^{\prime}(t)={\mathfrak{X}_{2}}_{\varphi(t)}$, that is

$\displaystyle\sum_{i=1}^{5}x^{*^{\prime}}_{i}(t)\frac{\partial}{\partial
x^{*}_{i}}+x^{*^{\prime}}(t)\frac{\partial}{\partial
x^{*}}+y^{*^{\prime}}(t)\frac{\partial}{\partial y^{*}}=$
$\displaystyle\qquad=\lambda x^{*}_{2}(t)\frac{\partial}{\partial
x^{*}_{2}}+\lambda x^{*}_{3}(t)\frac{\partial}{\partial
x^{*}_{3}}+(\lambda+1)x^{*}_{4}(t)\frac{\partial}{\partial
x^{*}_{4}}+(\lambda+1)x^{*}_{5}(t)\frac{\partial}{\partial x^{*}_{5}}$
$\displaystyle\Leftrightarrow$
$\displaystyle\begin{cases}x^{*^{\prime}}_{1}(t)=x^{*^{\prime}}(t)=y^{*^{\prime}}(t)=0\\\
x^{*^{\prime}}_{2}(t)=\lambda x^{*}_{2}(t)\\\ x^{*^{\prime}}_{3}(t)=\lambda
x^{*}_{3}(t)\\\ x^{*^{\prime}}_{4}(t)=(\lambda+1)x^{*}_{4}(t)\\\
x^{*^{\prime}}_{5}(t)=(\lambda+1)x^{*}_{5}(t).\end{cases}$ (3.3)

Combining with condition $F=\varphi(0)$, we obtain

$x^{*}_{1}=\alpha_{1},\,x^{*}_{2}=\alpha_{2}e^{\lambda
t},\,x^{*}_{3}=\alpha_{3}e^{\lambda
t},\,x^{*}_{4}=\alpha_{4}e^{(\lambda+1)t},\,x^{*}_{5}=\alpha_{5}e^{(\lambda+1)t},\,x^{*}=\alpha,\,y^{*}=\beta.$
(3.4)

Therefore, the flow of $\mathfrak{X}_{2}$ is

$\theta^{\mathfrak{X}_{2}}_{x}:F\mapsto\theta^{\mathfrak{X}_{2}}_{x}(F):=(\alpha_{1},\alpha_{2}e^{\lambda
x},\alpha_{3}e^{\lambda
x},\alpha_{4}e^{(\lambda+1)x},\alpha_{5}e^{(\lambda+1)x},\alpha,\beta).$

Next, we consider
$\mathfrak{X}_{3}:=(x^{*}_{2}+x^{*}_{3})\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{3}\frac{\partial}{\partial
x^{*}_{3}}+(x^{*}_{4}+x^{*}_{5})\frac{\partial}{\partial
x^{*}_{4}}+x^{*}_{5}\frac{\partial}{\partial x^{*}_{5}}$. For some positive
$\epsilon\in{\mathbb{R}}$, assume that

$\varphi:t\mapsto\varphi(t)=\bigl{(}x^{*}_{1}(t),x^{*}_{2}(t),x^{*}_{3}(t),x^{*}_{4}(t),x^{*}_{5}(t),x^{*}(t),y^{*}(t)\bigr{)};t\in(-\epsilon,\epsilon)$

is an integral curve of $\mathfrak{X}_{3}$ passing $F=\varphi(0)$. Then,
$\varphi^{\prime}(t)={\mathfrak{X}_{2}}_{\varphi(t)}$ which is equivalent to

$\displaystyle\sum_{i=1}^{5}x^{*^{\prime}}_{i}(t)\frac{\partial}{\partial
x^{*}_{i}}+x^{*^{\prime}}(t)\frac{\partial}{\partial
x^{*}}+y^{*^{\prime}}(t)\frac{\partial}{\partial y^{*}}=$
$\displaystyle\qquad=\left[x^{*}_{2}(t)+x^{*}_{3}(t)\right]\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{3}(t)\frac{\partial}{\partial
x^{*}_{3}}+\left[x^{*}_{4}(t)+x^{*}_{5}(t)\right]\frac{\partial}{\partial
x^{*}_{4}}+x^{*}_{5}(t)\frac{\partial}{\partial x^{*}_{5}}$
$\displaystyle\Leftrightarrow$
$\displaystyle\begin{cases}x^{*^{\prime}}_{1}(t)=x^{*^{\prime}}(t)=y^{*^{\prime}}(t)=0\\\
x^{*^{\prime}}_{2}(t)=x^{*}_{2}(t)+x^{*}_{3}(t)\\\
x^{*^{\prime}}_{3}(t)=x^{*}_{3}(t)\\\
x^{*^{\prime}}_{4}(t)=x^{*}_{4}(t)+x^{*}_{5}(t)\\\
x^{*^{\prime}}_{5}(t)=x^{*}_{5}(t).\end{cases}$ (3.5)

Combining with condition $F=\varphi(0)$, systemn (3) gives us

$\begin{array}[]{ll}&x^{*}_{1}=\alpha_{1},\;x^{*}_{2}=\alpha_{2}e^{t}+\alpha_{3}te^{t},\;x^{*}_{3}=\alpha_{3}e^{t},\\\
&x^{*}_{4}=\alpha_{4}e^{t}+\alpha_{5}ye^{t},\;x^{*}_{5}=\alpha_{5}e^{t},\;x^{*}=\alpha,\quad
y^{*}=\beta.\end{array}$

Hence, the flow of $\mathfrak{X}_{3}$ is

$\theta^{\mathfrak{X}_{3}}_{y}:F\mapsto\theta^{\mathfrak{X}_{3}}_{y}(F):=(\alpha_{1},\alpha_{2}e^{y}+\alpha_{3}ye^{y},\alpha_{3}e^{y},\alpha_{4}e^{y}+\alpha_{5}ye^{y},\alpha_{5}e^{y},\alpha,\beta).$

Now, we consider $\mathfrak{X}_{4}:=x^{*}_{4}\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{5}\frac{\partial}{\partial x^{*}_{3}}$. Assume that

$\varphi:t\mapsto\varphi(t)=\bigl{(}x^{*}_{1}(t),x^{*}_{2}(t),x^{*}_{3}(t),x^{*}_{4}(t),x^{*}_{5}(t),x^{*}(t),y^{*}(t)\bigr{)}$

is an integral curve of $\mathfrak{X}_{4}$ passing $F=\varphi(0)$, where
$t\in(-\epsilon,\epsilon)\subset\mathbb{R}$ for some positive
$\epsilon\in\mathbb{R}$. Then we have
$\varphi^{\prime}(t)={\mathfrak{X}_{4}}_{\varphi(t)}$ which means that

$\displaystyle\sum_{i=1}^{5}x^{*^{\prime}}_{i}(t)\frac{\partial}{\partial
x^{*}_{i}}+x^{*^{\prime}}(t)\frac{\partial}{\partial
x^{*}}+y^{*^{\prime}}(t)\frac{\partial}{\partial
y^{*}}=x^{*}_{4}(t)\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{5}(t)\frac{\partial}{\partial x^{*}_{3}}$
$\displaystyle\Leftrightarrow$
$\displaystyle\begin{cases}x^{*^{\prime}}_{1}(t)=x^{*^{\prime}}_{4}(t)=x^{*^{\prime}}_{5}(t)=x^{*^{\prime}}(t)=y^{*^{\prime}}(t)=0\\\
x^{*^{\prime}}_{2}(t)=x^{*}_{4}(t)\\\
x^{*^{\prime}}_{3}(t)=x^{*}_{5}(t).\end{cases}$ (3.6)

Since $F=\varphi(0)$, system (3) gives us

$\displaystyle
x^{*}_{1}=\alpha_{1},\;x^{*}_{2}=\alpha_{2}+\alpha_{4}t,\;x^{*}_{3}=\alpha_{3}+\alpha_{5}t,\;x^{*}_{4}=\alpha_{4},\;x^{*}_{5}=\alpha_{5},\;x^{*}=\alpha,\;y^{*}=\beta.$
(3.7)

Therefore, the flow of $\mathfrak{X}_{4}$ is

$\theta^{\mathfrak{X}_{4}}_{x_{1}}:F\mapsto\theta^{\mathfrak{X}_{4}}_{x_{1}}(F):=(\alpha_{1},\alpha_{2}+\alpha_{4}x_{1},\alpha_{3}+\alpha_{5}x_{1},\alpha_{4},\alpha_{5},\alpha,\beta).$

By setting
$\theta=\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}\circ\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}\circ\theta^{\mathfrak{X}_{4}}_{x_{1}}\circ\theta^{\mathfrak{X}_{3}}_{y}\circ\theta^{\mathfrak{X}_{2}}_{x}\circ\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}$,
we have

$\displaystyle\theta(F)$
$\displaystyle=\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}\circ\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}\circ\theta^{\mathfrak{X}_{4}}_{x_{1}}\circ\theta^{\mathfrak{X}_{3}}_{y}\circ\theta^{\mathfrak{X}_{2}}_{x}\circ\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}(F)$
$\displaystyle=(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})$

where $x^{*}_{1},x^{*},y^{*}\in\mathbb{R}$ and

$\begin{array}[]{l}x^{*}_{2}=\alpha_{2}e^{y}+\alpha_{4}x_{1}e^{y},\;x^{*}_{3}=\alpha_{3}e^{\lambda_{1}x+\lambda_{2}y}+\alpha_{5}x_{1}e^{\lambda_{1}x+\lambda_{2}y},\;x^{*}_{4}=\alpha_{4}e^{x+y},\;x^{*}_{5}=\alpha_{5}e^{\lambda_{1}x+x+\lambda_{2}y}.\end{array}$

By direct calculations, we get

$\begin{array}[]{ll}&x^{*}_{2}=\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}+\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{{x^{*}_{5}}^{\frac{\lambda}{1+\lambda}}e^{\frac{x^{*}_{4}}{(1+\lambda)x^{*}_{5}}}}{\alpha_{5}^{\frac{\lambda}{1+\lambda}}e^{\frac{\alpha_{4}}{(1+\lambda)\alpha_{5}}}}\\\
\Leftrightarrow&\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{1}{{x^{*}_{5}}^{\frac{\lambda}{1+\lambda}}e^{\frac{x^{*}_{4}}{(1+\lambda)x^{*}_{5}}}}=\left(\alpha_{2}-\frac{\alpha_{3}\alpha_{4}}{\alpha_{5}}\right)\frac{1}{\alpha_{5}^{\frac{\lambda}{1+\lambda}}e^{\frac{\alpha_{4}}{(1+\lambda)\alpha_{5}}}}\cdot\end{array}$

Hence,
$\left\\{\theta(F)\,|\,x^{*}_{1},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\in\mathbb{R};\alpha_{5}x^{*}_{5}>0\right\\}\equiv\Omega_{F}$,
i.e. $\Omega_{F}$ is a maximal connected integral submanifold corresponding to
$S_{G}$. Therefore, $S_{G}$ generates $\mathcal{F}_{G}$ and
$(V_{G},\mathcal{F}_{G})$ is a six-dimensional foliation for
$G=G_{12}^{\lambda}$.

Proof of Step 2. Similar to the case $G=G_{4}^{\lambda_{1},\lambda_{2}}$, we
also show that the foliation $(V_{G},\mathcal{F}_{G})$ is measurable in the
sense of Connes for $G=G_{12}^{\lambda}$. The proof is complete for the case
$G=G_{12}^{\lambda}$.

Finally, let $G=G_{13}^{\lambda}$. Proof of Step 1. For any
$F(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},,\alpha_{5},\alpha,\beta)\in\mathcal{G}^{*}$,
by Theorem 1, the generic K-orbit $\Omega_{F}$ belongs to $\mathcal{F}_{G}$ if
and only if $\alpha_{4}\alpha_{5}\neq 0$. Furthermore, if we denote by $v$ the
element
$(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in\mathcal{G}^{*}$
then

$\begin{array}[]{r}\Omega_{F}=\Bigg{\\{}v\in\mathcal{G}^{*}\,\bigg{|}\;\frac{x^{*}_{2}x^{*}_{5}-x^{*}_{3}x^{*}_{4}}{{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}}e^{\lambda\arctan\frac{x^{*}_{4}}{x^{*}_{5}}}=\frac{\alpha_{2}\alpha_{5}-\alpha_{3}\alpha_{4}}{{\alpha_{4}}^{2}+{\alpha_{5}}^{2}}e^{\lambda\arctan\frac{\alpha_{4}}{\alpha_{5}}}\Bigg{\\}}.\end{array}$

On the open submanifold $V_{G}=V_{3}$ as in (2.6), we consider the following
differential system $S_{G}$:

$\begin{cases}\mathfrak{X}_{1}:=\frac{\partial}{\partial x^{*}_{1}}\\\
\mathfrak{X}_{2}:=x^{*}_{2}\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{3}\frac{\partial}{\partial
x^{*}_{3}}+x^{*}_{4}\frac{\partial}{\partial
x^{*}_{4}}+x^{*}_{5}\frac{\partial}{\partial x^{*}_{5}}\\\
\mathfrak{X}_{3}:=x^{*}_{3}\frac{\partial}{\partial
x^{*}_{2}}-x^{*}_{2}\frac{\partial}{\partial x^{*}_{3}}+\lambda
x^{*}_{5}\frac{\partial}{\partial x^{*}_{4}}-\lambda
x^{*}_{4}\frac{\partial}{\partial x^{*}_{5}}\\\
\mathfrak{X}_{4}:=x^{*}_{4}\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{5}\frac{\partial}{\partial x^{*}_{3}}\\\
\mathfrak{X}_{5}:=\frac{\partial}{\partial x^{*}}\\\
\mathfrak{X}_{6}:=\frac{\partial}{\partial y^{*}}\cdot\end{cases}$

Obviously, $\operatorname{rank}(S_{G})=6$ and all
$\mathfrak{X}_{i}\,(i=1,\ldots,6)$ are smooth over $V_{G}$. Moreover, we will
show that $S_{G}$ generates $\mathcal{F}_{G}$, i.e. each K-orbit $\Omega$ from
$\mathcal{F}_{G}$ is a maximal connected integral submanifold of $S_{G}$.

First, we consider $\mathfrak{X}_{1},\mathfrak{X}_{5},\mathfrak{X}_{6}$.
Clearly, their flows are determined as follows

$\begin{array}[]{ll}\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}:&F\mapsto\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}(F):=(x^{*}_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha,\beta)\\\
\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}:&F\mapsto\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}(F):=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},x^{*},\beta)\\\
\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}:&F\mapsto\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}(F):=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha,y^{*}).\end{array}$

Next, we consider $\mathfrak{X}_{2}:=x^{*}_{2}\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{3}\frac{\partial}{\partial
x^{*}_{3}}+x^{*}_{4}\frac{\partial}{\partial
x^{*}_{4}}+x^{*}_{5}\frac{\partial}{\partial x^{*}_{5}}$. For some positive
$\epsilon\in{\mathbb{R}}$, assume that

$\varphi:t\mapsto\varphi(t)=\bigl{(}x^{*}_{1}(t),x^{*}_{2}(t),x^{*}_{3}(t),x^{*}_{4}(t),x^{*}_{5}(t),x^{*}(t),y^{*}(t)\bigr{)};t\in(-\epsilon,\epsilon)$

is an integral curve of $\mathfrak{X}_{2}$ passing $F=\varphi(0)$. Hence, we
have $\varphi^{\prime}(t)={\mathfrak{X}_{2}}_{\varphi(t)}$, and this means
that

$\displaystyle\sum_{i=1}^{5}x^{*^{\prime}}_{i}(t)\frac{\partial}{\partial
x^{*}_{i}}+x^{*^{\prime}}(t)\frac{\partial}{\partial
x^{*}}+y^{*^{\prime}}(t)\frac{\partial}{\partial
y^{*}}=x^{*}_{2}(t)\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{3}(t)\frac{\partial}{\partial
x^{*}_{3}}+x^{*}_{4}(t)\frac{\partial}{\partial
x^{*}_{4}}+x^{*}_{5}(t)\frac{\partial}{\partial x^{*}_{5}}$
$\displaystyle\Leftrightarrow$
$\displaystyle\begin{cases}x^{*^{\prime}}_{1}(t)=x^{*^{\prime}}(t)=y^{*^{\prime}}(t)=0\\\
x^{*^{\prime}}_{2}(t)=x^{*}_{2}(t)\\\ x^{*^{\prime}}_{3}(t)=x^{*}_{3}(t)\\\
x^{*^{\prime}}_{4}(t)=x^{*}_{4}(t)\\\
x^{*^{\prime}}_{5}(t)=x^{*}_{5}(t).\end{cases}$ (3.8)

Combining with condition $F=\varphi(0)$, we obtain

$x^{*}_{1}=\alpha_{1},\,x^{*}_{2}=\alpha_{2}e^{t},\,x^{*}_{3}=\alpha_{3}e^{t},\,x^{*}_{4}=\alpha_{4}e^{t},\,x^{*}_{5}=\alpha_{5}e^{t},\,x^{*}=\alpha,\,y^{*}=\beta.$

Therefore, the flow of $\mathfrak{X}_{2}$ is

$\theta^{\mathfrak{X}_{2}}_{x}:F\mapsto\theta^{\mathfrak{X}_{2}}_{x}(F):=(\alpha_{1},\alpha_{2},\alpha_{3}e^{x},\alpha_{4}e^{x},\alpha_{5}e^{x},\alpha,\beta).$

Next, we consider $\mathfrak{X}_{3}:=x^{*}_{3}\frac{\partial}{\partial
x^{*}_{2}}-x^{*}_{2}\frac{\partial}{\partial x^{*}_{3}}+\lambda
x^{*}_{5}\frac{\partial}{\partial x^{*}_{4}}-\lambda
x^{*}_{4}\frac{\partial}{\partial x^{*}_{5}}$. For some positive
$\epsilon\in{\mathbb{R}}$, assume that

$\varphi:t\mapsto\varphi(t)=\bigl{(}x^{*}_{1}(t),x^{*}_{2}(t),x^{*}_{3}(t),x^{*}_{4}(t),x^{*}_{5}(t),x^{*}(t),y^{*}(t)\bigr{)};t\in(-\epsilon,\epsilon)$

is an integral curve of $\mathfrak{X}_{3}$ passing $F=\varphi(0)$. Hence, we
have $\varphi^{\prime}(t)={\mathfrak{X}_{3}}_{\varphi(t)}$, that is

$\displaystyle\sum_{i=1}^{5}x^{*^{\prime}}_{i}(t)\frac{\partial}{\partial
x^{*}_{i}}+x^{*^{\prime}}(t)\frac{\partial}{\partial
x^{*}}+y^{*^{\prime}}(t)\frac{\partial}{\partial
y^{*}}=x^{*}_{3}(t)\frac{\partial}{\partial
x^{*}_{2}}-x^{*}_{2}(t)\frac{\partial}{\partial x^{*}_{3}}+\lambda
x^{*}_{5}(t)\frac{\partial}{\partial x^{*}_{4}}-\lambda
x^{*}_{4}(t)\frac{\partial}{\partial x^{*}_{5}}$
$\displaystyle\Leftrightarrow$
$\displaystyle\begin{cases}x^{*^{\prime}}_{1}(t)=x^{*^{\prime}}(t)=y^{*^{\prime}}(t)=0\\\
x^{*^{\prime}}_{2}(t)=x^{*}_{3}(t)\\\ x^{*^{\prime}}_{3}(t)=-x^{*}_{2}(t)\\\
x^{*^{\prime}}_{4}(t)=\lambda x^{*}_{5}(t)\\\ x^{*^{\prime}}_{5}(t)=-\lambda
x^{*}_{4}(t).\end{cases}$ (3.9)

Combining with condition $F=\varphi(0)$, system (3) gives us

$\begin{array}[]{ll}&x^{*}_{1}=\alpha_{1},\\\
&x^{*}_{2}=\alpha_{2}\cos{t}+\alpha_{3}\sin{t},\\\
&x^{*}_{3}=-\alpha_{2}\sin{t}+\alpha_{3}\cos{t},\\\
&x^{*}_{4}=(\alpha_{4}\cos{t}+\alpha_{5}\sin{t})e^{\lambda t},\\\
&x^{*}_{5}=(-\alpha_{4}\sin{t}+\alpha_{5}\cos{t})e^{\lambda t},\\\
&x^{*}=\alpha,\quad y^{*}=\beta.\end{array}$

Hence, the flow of $\mathfrak{X}_{3}$ is

$\begin{array}[]{l}\theta^{\mathfrak{X}_{3}}_{y}:F\mapsto\theta^{\mathfrak{X}_{3}}_{y}(F):=\big{(}\alpha_{1},\alpha_{2}\cos{y}+\alpha_{3}\sin{y},-\alpha_{2}\sin{y}+\alpha_{3}\cos{y},\\\
\hskip 142.26378pt(\alpha_{4}\cos{y}+\alpha_{5}\sin{y})e^{\lambda
y},(-\alpha_{4}\sin{y}+\alpha_{5}\cos{y})e^{\lambda
y},\alpha,\beta\big{)}.\end{array}$

Now, we consider $\mathfrak{X}_{4}:=x^{*}_{4}\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{5}\frac{\partial}{\partial x^{*}_{3}}$. Assume that

$\varphi:t\mapsto\varphi(t)=\bigl{(}x^{*}_{1}(t),x^{*}_{2}(t),x^{*}_{3}(t),x^{*}_{4}(t),x^{*}_{5}(t),x^{*}(t),y^{*}(t)\bigr{)}$

is an integral curve of $\mathfrak{X}_{4}$ passing $F=\varphi(0)$, where
$t\in(-\epsilon,\epsilon)\subset\mathbb{R}$ for some positive
$\epsilon\in\mathbb{R}$. Then,
$\varphi^{\prime}(t)={\mathfrak{X}_{4}}_{\varphi(t)}$ which is equivalent to

$\displaystyle\sum_{i=1}^{5}x^{*^{\prime}}_{i}(t)\frac{\partial}{\partial
x^{*}_{i}}+x^{*^{\prime}}(t)\frac{\partial}{\partial
x^{*}}+y^{*^{\prime}}(t)\frac{\partial}{\partial
y^{*}}=x^{*}_{4}(t)\frac{\partial}{\partial
x^{*}_{2}}+x^{*}_{5}(t)\frac{\partial}{\partial x^{*}_{3}}$
$\displaystyle\Leftrightarrow$
$\displaystyle\begin{cases}x^{*^{\prime}}_{1}(t)=x^{*^{\prime}}_{4}(t)=x^{*^{\prime}}_{5}(t)=x^{*^{\prime}}(t)=y^{*^{\prime}}(t)=0\\\
x^{*^{\prime}}_{2}(t)=x^{*}_{4}(t)\\\
x^{*^{\prime}}_{3}(t)=x^{*}_{5}(t).\end{cases}$ (3.10)

Since $F=\varphi(0)$, system (3) gives us

$\displaystyle
x^{*}_{1}=\alpha_{1},\;x^{*}_{2}=\alpha_{2}+\alpha_{4}t,\;x^{*}_{3}=\alpha_{3}+\alpha_{5}t,\;x^{*}_{4}=\alpha_{4},\;x^{*}_{5}=\alpha_{5},\;x^{*}=\alpha,\;y^{*}=\beta.$

Thus, the flow of $\mathfrak{X}_{4}$ is

$\theta^{\mathfrak{X}_{4}}_{x_{1}}:F\mapsto\theta^{\mathfrak{X}_{4}}_{x_{1}}(F):=(\alpha_{1},\alpha_{2}+\alpha_{4}x_{1},\alpha_{3}+\alpha_{5}x_{1},\alpha_{4},\alpha_{5},\alpha,\beta).$

By setting
$\theta=\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}\circ\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}\circ\theta^{\mathfrak{X}_{4}}_{x_{1}}\circ\theta^{\mathfrak{X}_{3}}_{y}\circ\theta^{\mathfrak{X}_{2}}_{x}\circ\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}$,
we have

$\begin{array}[]{rl}\theta(F)=&\theta^{\mathfrak{X}_{6}}_{y^{*}-\beta}\circ\theta^{\mathfrak{X}_{5}}_{x^{*}-\alpha}\circ\theta^{\mathfrak{X}_{4}}_{x_{1}}\circ\theta^{\mathfrak{X}_{3}}_{y}\circ\theta^{\mathfrak{X}_{2}}_{x}\circ\theta^{\mathfrak{X}_{1}}_{x^{*}_{1}-\alpha_{1}}(F)\\\
=&(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*}).\end{array}$

where $x^{*}_{1},x^{*},y^{*}\in\mathbb{R}$ and

$\begin{array}[]{ll}x^{*}_{2}=\alpha_{2}e^{x}\cos(y)+\alpha_{3}e^{x}\sin(y)+\alpha_{4}x_{1}e^{x}\cos(y)+\alpha_{5}x_{1}e^{x}\sin(y),\\\
x^{*}_{3}=-\alpha_{2}e^{x}\sin(y)+\alpha_{3}e^{x}\cos(y)-\alpha_{4}x_{1}e^{x}\sin(y)+\alpha_{5}x_{1}e^{x}\cos(y),\\\
x^{*}_{4}=\alpha_{4}e^{x+\lambda y}\cos(y)+\alpha_{5}e^{x+\lambda
y}\sin(y),\\\ x^{*}_{5}=-\alpha_{4}e^{x+\lambda
y}\sin(y)+\alpha_{5}e^{x+\lambda y}\cos(y).\end{array}$

By direct calculations, we get

$\begin{array}[]{ll}&x^{*}_{2}=\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}+\frac{\alpha_{2}\alpha_{5}-\alpha_{3}\alpha_{4}}{{\alpha_{4}}^{2}+{\alpha_{5}}^{2}}\frac{e^{\lambda\arctan\frac{\alpha_{4}}{\alpha_{5}}}\left({x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}\right)}{x^{*}_{5}e^{\lambda\arctan\frac{x^{*}_{4}}{x^{*}_{5}}}}\\\
\Leftrightarrow&\frac{x^{*}_{2}x^{*}_{5}-x^{*}_{3}x^{*}_{4}}{{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}}e^{\lambda\arctan\frac{x^{*}_{4}}{x^{*}_{5}}}=\frac{\alpha_{2}\alpha_{5}-\alpha_{3}\alpha_{4}}{{\alpha_{4}}^{2}+{\alpha_{5}}^{2}}e^{\lambda\arctan\frac{\alpha_{4}}{\alpha_{5}}}.\end{array}$

Hence,
$\left\\{\theta(F)\,|\,x^{*}_{1},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\in\mathbb{R};{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}\neq
0\right\\}\equiv\Omega_{F}$, i.e. $\Omega_{F}$ is a maximal connected integral
submanifold of $S_{G}$. Therefore, $S_{G}$ generates $\mathcal{F}_{G}$ and
$(V_{G},\mathcal{F}_{G})$ is a six-dimensional foliation for
$G=G_{13}^{\lambda}$.

Proof of Step 2. Similar to the case $G=G_{4}^{\lambda_{1},\lambda_{2}}$, we
also show that the foliation $(V_{G},\mathcal{F}_{G})$ is measurable in the
sense of Connes for $G=G_{13}^{\lambda}$.

For the remaining cases, the proof is entirely analogous. Here, we only give
the systems $S_{G}$ in Table LABEL:tab3 in which the disappearance of
coefficients means they are zeros.

Table 5: The differential systems $S_{G}$ $S_{G}$ |  $\mathfrak{X}_{i}$ | Coefficients
---|---|---
$\frac{\partial}{\partial x^{*}_{1}}$ |  $\frac{\partial}{\partial x^{*}_{2}}$ |  $\frac{\partial}{\partial x^{*}_{3}}$ |  $\frac{\partial}{\partial x^{*}_{4}}$ |  $\frac{\partial}{\partial x^{*}_{5}}$ |  $\frac{\partial}{\partial x^{*}}$ |  $\frac{\partial}{\partial y^{*}}$
$S_{G_{1}^{\lambda}}$ |  $\mathfrak{X}_{1}$ |  1 |  |  |  |  |  |
$\mathfrak{X}_{2}$ |  |  |  $x_{3}^{*}$ |  |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{3}$ |  |  $-x_{2}^{*}$ |  |  |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{4}$ |  |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |  |  |
$\mathfrak{X}_{5}$ |  |  |  |  |  |  1 |
$\mathfrak{X}_{6}$ |  |  |  |  |  |  |  1
$S_{G_{5}}$ |  $\mathfrak{X}_{1}$ |  1 |  |  |  |  |  |
$\mathfrak{X}_{2}$ |  |  |  $x_{3}^{*}$ |  |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{3}$ |  |  $x_{2}^{*}$ |  |  $2x_{4}^{*}$ |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{4}$ |  |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |  |  |
$\mathfrak{X}_{5}$ |  |  |  |  |  |  1 |
$\mathfrak{X}_{6}$ |  |  |  |  |  |  |  1
$S_{G_{6}^{\lambda}}$ |  $\mathfrak{X}_{1}$ |  1 |  |  |  |  |  |
$\mathfrak{X}_{2}$ |  |  |  $x_{3}^{*}$ |  |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{3}$ |  |  $x_{2}^{*}$ |  $\lambda x_{3}^{*}$ |  $2x_{4}^{*}$ |  $(\lambda+1)x_{5}^{*}$ |  |
$\mathfrak{X}_{4}$ |  |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |  |  |
$\mathfrak{X}_{5}$ |  |  |  |  |  |  1 |
$\mathfrak{X}_{6}$ |  |  |  |  |  |  |  1
$S_{G_{7}}$ |  $\mathfrak{X}_{1}$ |  1 |  |  |  |  |  |
$\mathfrak{X}_{2}$ |  |  $x_{2}^{*}$ |  $x_{3}^{*}$ |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{3}$ |  |  $x_{2}^{*}+x_{5}^{*}$ |  |  $2x_{4}^{*}$ |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{4}$ |  |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |  |  |
$\mathfrak{X}_{5}$ |  |  |  |  |  |  1 |
$\mathfrak{X}_{6}$ |  |  |  |  |  |  |  1
$S_{G_{8}^{\lambda}}$ |  $\mathfrak{X}_{1}$ |  1 |  |  |  |  |  |
$\mathfrak{X}_{2}$ |  |  $(\lambda+1)x_{2}^{*}$ |  $\lambda x_{3}^{*}$ |  $(\lambda+2)x_{4}^{*}$ |  $(\lambda+1)x_{5}^{*}$ |  |
$\mathfrak{X}_{3}$ |  |  $x_{2}^{*}+x_{5}^{*}$ |  $x_{3}^{*}$ |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{4}$ |  |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |  |  |
$\mathfrak{X}_{5}$ |  |  |  |  |  |  1 |
$\mathfrak{X}_{6}$ |  |  |  |  |  |  |  1
$S_{G_{11}}$ |  $\mathfrak{X}_{1}$ |  1 |  |  |  |  |  |
$\mathfrak{X}_{2}$ |  |  $x_{2}^{*}$ |  $x_{3}^{*}$ |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{3}$ |  |  $x_{3}^{*}$ |  |  $x_{4}^{*}+x_{5}^{*}$ |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{4}$ |  |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |  |  |
$\mathfrak{X}_{5}$ |  |  |  |  |  |  1 |
$\mathfrak{X}_{6}$ |  |  |  |  |  |  |  1
$S_{G_{14}^{\lambda_{1}\lambda_{2}}}$ |  $\mathfrak{X}_{1}$ |  1 |  |  |  |  |  |
$\mathfrak{X}_{2}$ |  |  $\lambda_{1}x_{2}^{*}$ |  $\lambda_{1}x_{3}^{*}$ |  $(1+\lambda_{1})x_{4}^{*}$ |  $(1+\lambda_{1})x_{5}^{*}$ |  |
$\mathfrak{X}_{3}$ |  |  $\lambda_{2}x_{2}^{*}+x_{3}^{*}$ |  $-x_{2}^{*}+\lambda_{2}x_{3}^{*}$ |  $\lambda_{2}x_{4}^{*}-x_{5}^{*}$ |  $-x_{4}^{*}+\lambda_{2}x_{5}^{*}$ |  |
$\mathfrak{X}_{4}$ |  |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |  |  |
$\mathfrak{X}_{5}$ |  |  |  |  |  |  1 |
$\mathfrak{X}_{6}$ |  |  |  |  |  |  |  1
$S_{G_{15}}$ |  $\mathfrak{X}_{1}$ |  1 |  |  |  |  |  |
$\mathfrak{X}_{2}$ |  |  $x_{3}^{*}$ |  $-x_{2}^{*}$ |  $x_{5}^{*}$ |  $-x_{4}^{*}$ |  |
$\mathfrak{X}_{3}$ |  |  $x_{2}^{*}+x_{5}^{*}$ |  $x_{3}^{*}-x_{4}^{*}$ |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{4}$ |  |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |  |  |
$\mathfrak{X}_{5}$ |  |  |  |  |  |  1 |
$\mathfrak{X}_{6}$ |  |  |  |  |  |  |  1
$S_{G_{16}^{\lambda}}$ |  $\mathfrak{X}_{1}$ |  1 |  |  |  |  |  |
$\mathfrak{X}_{2}$ |  |  $x_{3}^{*}+x_{5}^{*}$ |  $-x_{2}^{*}$ |  $x_{5}^{*}$ |  $-x_{4}^{*}$ |  |
$\mathfrak{X}_{3}$ |  |  $x_{2}^{*}+\lambda x_{5}^{*}$ |  $x_{3}^{*}-\lambda x_{4}^{*}$ |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |
$\mathfrak{X}_{4}$ |  |  $x_{4}^{*}$ |  $x_{5}^{*}$ |  |  |  |
$\mathfrak{X}_{5}$ |  |  |  |  |  |  1 |
$\mathfrak{X}_{6}$ |  |  |  |  |  |  |  1

The proof of Theorem 2 is complete. ∎

######  Definition 3.1.

For each $G\in\mathcal{A}$, the foliation $(V_{G},\mathcal{F}_{G})$ is called
the _generalized MD-foliation_ (for short, _GMD-foliation_) associated to $G$.

######  Remark 3.2.

By Remark 2.4 and Theorem 2, we have exactly sixteen families of measurable
GMD-foliations $\\{(V_{G},\mathcal{F}_{G})/G\in\mathcal{A}\\}$. However, there
are only three types of foliated manifolds $V_{G}$. Namely, we have

  * 1.

For all $G\in\mathcal{A}_{1}\setminus\\{G_{12}^{\lambda}\\}$, $V_{G}\equiv
V_{1}$ as in (2.1);

  * 2.

For $G=G_{12}^{\lambda}$, $V_{G}\equiv V_{2}$ as in (2.4);

  * 3.

For all $G\in\mathcal{A}_{2}\big{\\}}$, $V_{G}\equiv V_{3}$ as in (2.6).

It is this observation that leads us to the following result about the
topological classification of sixteen families of GMD-foliations. Namely, we
have the following theorem.

######  Theorem 3.

The topology of GMD-foliations has the following properties.

  1. 1.

There exist exactly three topological types of sixteen families of considered
GMD-foliations as follows:

    1. (a)

The Type $\mathcal{F}_{1}$: This type contains eleven families of foliated
manifolds from
$\big{\\{}(V_{G},\mathcal{F}_{G})/G\in\mathcal{A}_{1}\setminus\\{G_{12}^{\lambda}\\}\big{\\}}$
on the same foliated manifold $V_{G}\equiv V_{1}$ as in (2.1);

    2. (b)

The Type $\mathcal{F}_{2}$: This type contains only one familiy of foliated
manifolds from
$\big{\\{}\big{(}V_{2},\mathcal{F}_{G_{12}^{\lambda}}\big{)}\big{\\}}$ on the
same foliated manifold $V_{G}\equiv V_{2}$ as in (2.4);

    3. (c)

The Type $\mathcal{F}_{3}$: This type contains four families of foliated
manifolds from $\big{\\{}(V_{G},\mathcal{F}_{G})/G\in\mathcal{A}_{2}\big{\\}}$
on the same foliated manifold $V_{G}\equiv V_{3}$ as in (2.6).

  2. 2.

Furthermore, we have:

    1. (a)

All GMD-foliations of type $\mathcal{F}_{1}$ are trivial fibrations with
connected fibers on the disjoint union of four copies of the real line
$\mathbb{R}$.

    2. (b)

The GMD-foliation of type $\mathcal{F}_{2}$ is a trivial fibration with
connected fibers on the disjoint union of two copies of the real line
$\mathbb{R}$.

    3. (c)

All GMD-foliations of type $\mathcal{F}_{3}$ are trivial fibrations with
connected fibers on a copy of the real line $\mathbb{R}$.

###### Proof.

Recall that two foliations $\mathcal{F}$ and $\mathcal{F}^{\prime}$ on the
same foliated manifold $V$ are _topologically equivalent_ if there exists a
homeomorphism $h\colon V\to V$ which sends leaves of $\mathcal{F}$ onto those
of $\mathcal{F}^{\prime}$.

  1. 1.

Note that the assertion in Item (1a) of Theorem 3 for cases $G_{2}$, $G_{3}$,
$G_{9}$ and $G^{\lambda}_{10}$ had been proved in [17]. The proof is quite
analogous for remaining cases. First, we prove the topological classification
of considered GMD-foliations in the first assertion of Theorem 3.

    1. (a)

Consider maps
$h_{1(\lambda)},h_{2(\lambda_{1}\lambda_{2})},h_{3(\lambda)},h_{4},h_{5},h_{6}$
from $V_{1}$ to $V_{1}$ which are defined as follows:

$\begin{array}[]{l}h_{1(\lambda)}(v):=\left(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}},x^{*}_{5},x^{*},y^{*}\right),\\\
h_{2(\lambda_{1}\lambda_{2})}(v):=\left(x^{*}_{1},\frac{x^{*}_{2}{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}},\frac{x^{*}_{3}{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right),\\\
h_{3}(v):=\left(x^{*}_{1},\left(x^{*}_{2}+\ln|x^{*}_{4}|\right)x^{*}_{5},\left(x^{*}_{3}+\frac{x^{*}_{5}\ln|x^{*}_{5}|}{x^{*}_{4}}\right)x^{*}_{5},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right),\\\
h_{4(\lambda)}(v):=\left(x^{*}_{1},\left(x^{*}_{2}-(1+\lambda)\ln|x^{*}_{4}|\right)x^{*}_{5},\left(x^{*}_{3}-\frac{(2+\lambda)x^{*}_{5}\ln|x^{*}_{5}|}{x^{*}_{4}}\right)x^{*}_{5},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right),\\\
h_{5}(v):=\left(x^{*}_{1},\frac{x^{*}_{2}x^{*}_{5}}{e^{\frac{x^{*}_{4}}{x_{5}}}},\frac{x^{*}_{3}x^{*}_{5}}{e^{\frac{x^{*}_{4}}{x_{5}}}},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right),\\\
h_{6}(v):=\left(x^{*}_{1},\frac{x^{*}_{2}}{\sqrt{|x^{*}_{4}|}},\frac{x^{*}_{3}}{\sqrt{|x^{*}_{4}|}},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right)\end{array}$

where $v:=(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in
V_{1}$. It is clear that all of above maps are homeomorphisms. Now, we take an
arbitrary leaf $L$ of $(V_{1},\mathcal{F}_{G_{2}})$. Without loss of
generality, assume that $L\subset{V_{1}}_{++}\subset V_{1}$ (see (2.2) and
(2.3)), i.e. $L$ is determined as follows

$L=\left\\{v\in
V_{1}\,\Big{|}\,x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}=c;\,\,x^{*}_{4}>0,\,x^{*}_{5}>0\right\\}$

where $c\in\mathbb{R}$ is some constant. For
$\left(V_{1},\mathcal{F}_{G_{4}^{\lambda_{1}\lambda_{2}}}\right)$, we consider
the leaf $\tilde{L}\subset{V_{1}}_{++}\subset V_{1}$ which is determined as
follows

$\tilde{L}=\left\\{\tilde{v}\in
V_{1}\,\Big{|}\left(\tilde{x}_{2}-\frac{\tilde{x}_{3}\tilde{x}_{4}}{\tilde{x}_{5}}\right)\dfrac{\tilde{x}_{4}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{\tilde{x}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}=c;\;\widetilde{x}_{4}>0,\,\widetilde{x}_{5}>0\right\\}$

where
$\tilde{v}=(\tilde{x}_{1},\tilde{x}_{2},\tilde{x}_{3},\tilde{x}_{4},\tilde{x}_{5},\tilde{x},\tilde{y})\in
V_{1}$. By the formula of the homeomorphism $h_{2(\lambda_{1}\lambda_{2})}$,
the fact that $h_{2(\lambda_{1},\lambda_{2})}(v)=\tilde{v}$ is equivalent to

$\begin{array}[]{rl}&\tilde{v}=\left(x^{*}_{1},\frac{x^{*}_{2}{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}},\frac{x^{*}_{3}{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right)\\\
\Leftrightarrow&\begin{cases}\tilde{x}_{1}=x^{*}_{1}\\\
\tilde{x}_{2}=\frac{x^{*}_{2}{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}\\\
\tilde{x}_{3}=\frac{x^{*}_{3}{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}\\\
\tilde{x}_{4}=x^{*}_{4},\,\tilde{x}_{5}=x^{*}_{5},\,\tilde{x}=x^{*},\,\tilde{y}=y^{*}.\end{cases}\end{array}$

Therefore

$\begin{array}[]{l l l l}v\in
L&\Leftrightarrow&x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}=c;&x^{*}_{4}>0,\,x^{*}_{5}>0\\\
&\Leftrightarrow&\left(\tilde{x}_{2}-\frac{\tilde{x}_{3}\tilde{x}_{4}}{\tilde{x}_{5}}\right)\frac{\tilde{x}_{4}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}}{{\widetilde{x}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}=c;&\tilde{x}_{4}>0,\,\tilde{x}_{5}>0\\\
&\Leftrightarrow&h_{2(\lambda_{1}\lambda_{2})}(v)=\tilde{v}\in\tilde{L}\end{array}$

i.e. $h_{2(\lambda_{1}\lambda_{2})}(L)=\tilde{L}$ for $L\subset{V_{1}}_{++}$.
Similarly, $h_{2(\lambda_{1}\lambda_{2})}(L)=\tilde{L}$ for $L$ in
${V_{1}}_{-+}$, ${V_{1}}_{--}$, ${V_{1}}_{+-}$. This means that
$h_{2(\lambda_{1}\lambda_{2})}$ sends leaves of $\mathcal{F}_{G_{2}}$ onto
those of $\mathcal{F}_{G_{4}^{\lambda_{1}\lambda_{2}}}$. Therefore,
$\left(V_{1},\mathcal{F}_{G_{2}}\right)$ and
$\left(V_{1},\mathcal{F}_{G_{4}^{\lambda_{1}\lambda_{2}}}\right)$ are
topological equivalent.

In the same manner, we can verify that $h_{1(\lambda)}$, $h_{3}$,
$h_{4(\lambda)}$ and $h_{5}$ send leaves of $\mathcal{F}_{G_{2}}$ onto those
of $\mathcal{F}_{G^{\lambda}_{1}}$, $\mathcal{F}_{G_{7}}$,
$\mathcal{F}_{G_{8}^{\lambda}}$, $\mathcal{F}_{G_{11}}$, respectively.
Besides, $h_{6}$ sends leaves of $\mathcal{F}_{G_{3}}$ onto those of
$\mathcal{F}_{G_{5}}$ and $\mathcal{F}_{G_{6}^{\lambda}}$. Finally, combining
with [17, Theorem 2.3], all these foliations are topologically equivalent.
This type is denoted by $\mathcal{F}_{1}$.

    2. (b)

By direct calculations, the map $h_{7(\lambda\neq-1)}\colon
V_{2}\cong\mathbb{R}^{4}\times\left(\mathbb{R}\setminus\\{0\\}\right)\times\mathbb{R}^{2}\to
V_{2}$ defined by

$h_{7(\lambda\neq-1)}(v):=\left(x^{*}_{1},\frac{x^{*}_{2}{x^{*}_{5}}^{\frac{\lambda}{1+\lambda}}}{e^{\frac{\lambda
x^{*}_{4}}{(1+\lambda)x^{*}_{5}}}},\frac{x^{*}_{3}{x^{*}_{5}}^{\frac{\lambda}{1+\lambda}}}{e^{\frac{\lambda
x^{*}_{4}}{(1+\lambda)x^{*}_{5}}}},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right)$

is a homeomorphism which sends leaves of $\mathcal{F}_{G_{12}^{0}}$ onto those
of $\mathcal{F}_{G_{12}^{\lambda}}$.

In fact, take an arbitrary leaf $L$ of $\mathcal{F}_{G_{12}^{0}}$. Without
loss of generality, assume that $L\subset{V_{2}}_{+}\subset V_{2}$ as in (2.6)
and (2.5), i.e. $L$ is determined as follows

$L=\left\\{v\in
V_{2}\,\Big{|}\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{1}{e^{\frac{x^{*}_{4}}{x^{*}_{5}}}}=c;\;x^{*}_{5}>0\right\\}$

where $c\in\mathbb{R}$ is some constant. For $\mathcal{F}_{G_{12}^{\lambda}}$,
we consider the leaf $\tilde{L}\subset{V_{2}}_{+}\subset V_{2}$ as follows

$\tilde{L}=\left\\{\tilde{v}\in
V_{2}\,\Big{|}\left(\tilde{x}_{2}-\frac{\tilde{x}_{3}\tilde{x}_{4}}{\tilde{x}_{5}}\right)\frac{1}{{\tilde{x}_{5}}^{\frac{\lambda}{1+\lambda}}e^{\frac{{\tilde{x}}_{4}}{(1+\lambda)\tilde{x}_{5}}}}=c;\,\,\,\tilde{x}_{5}>0\right\\}$

where
$\tilde{v}=(\tilde{x}_{1},\tilde{x}_{2},\tilde{x}_{3},\tilde{x}_{4},\tilde{x}_{5},\tilde{x},\tilde{y})\in
V_{2}$. By the formula of $h_{7(\lambda\neq-1)}$, it is plain that
$h_{7(\lambda\neq-1)}(v)=\tilde{v}$ is equivalent to

$\begin{array}[]{rl}&\tilde{v}=\left(x^{*}_{1},\frac{x^{*}_{2}{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}},\frac{x^{*}_{3}{x^{*}_{5}}^{\frac{1}{\lambda_{2}-\lambda_{1}-1}}}{{x^{*}_{4}}^{\frac{1+\lambda_{1}}{\lambda_{2}-\lambda_{1}-1}}},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right)\\\
\Leftrightarrow&\begin{cases}\tilde{x}_{1}=x^{*}_{1}\\\
\tilde{x}_{2}=\frac{x^{*}_{2}{x^{*}_{5}}^{\frac{\lambda}{1+\lambda}}}{e^{\frac{\lambda
x^{*}_{4}}{(1+\lambda)x^{*}_{5}}}}\\\
\tilde{x}_{3}=\frac{x^{*}_{3}{x^{*}_{5}}^{\frac{\lambda}{1+\lambda}}}{e^{\frac{\lambda
x^{*}_{4}}{(1+\lambda)x^{*}_{5}}}}\\\
\tilde{x}_{4}=x^{*}_{4},\,\tilde{x}_{5}=x^{*}_{5},\,\tilde{x}=x^{*},\,\tilde{y}=y^{*}.\end{cases}\end{array}$

Therefore

$\begin{array}[]{l l l l}v\in
L&\Leftrightarrow&\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{1}{e^{\frac{x^{*}_{4}}{x^{*}_{5}}}}=c;&x^{*}_{5}>0\\\
&\Leftrightarrow&\left(\tilde{x}_{2}-\frac{\tilde{x}_{3}\tilde{x}_{4}}{\tilde{x}_{5}}\right)\frac{1}{{\tilde{x}_{5}}^{\frac{\lambda}{1+\lambda}}e^{\frac{{\tilde{x}}_{4}}{(1+\lambda)\tilde{x}_{5}}}}=c;&\tilde{x}_{5}>0\\\
&\Leftrightarrow&h_{7(\lambda\neq-1)}(v)=\tilde{v}\in\tilde{L}\end{array}$

i.e. $h_{7(\lambda\neq-1)}(L)=\tilde{L}$ for $L\subset{V_{2}}_{+}$. Similarly,
$h_{7(\lambda\neq-1)}(L)=\tilde{L}$ for $L$ in ${V_{2}}_{-}$. Since
$h_{7(\lambda\neq-1)}$ sends leaves of $\mathcal{F}_{G_{12}^{0}}$ onto those
of $\mathcal{F}_{G_{12}^{\lambda}}$, all foliations
$\left(V_{2},\mathcal{F}_{G_{12}^{\lambda}}\right)$ are topologically
equivelant. This types is denoted by $\mathcal{F}_{2}$.

    3. (c)

Similarly, consider homeomorphisms of $V_{3}$ as follows:

$\begin{array}[]{l}h_{8(\lambda)}(v):=\begin{cases}\left(x^{*}_{1},\frac{x^{*}_{2}}{e^{\lambda
a}},\frac{x^{*}_{3}}{e^{\lambda
a}},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right),&x^{*}_{4}x^{*}_{5}\neq 0\\\
(x^{*}_{1},x^{*}_{2},x^{*}_{3},0,x^{*}_{5},x^{*},y^{*}),&x^{*}_{4}=0,x^{*}_{5}\neq
0\\\ (x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},0,x^{*},y^{*}),&x^{*}_{4}\neq
0,x^{*}_{5}=0\end{cases}\\\
h_{9(\lambda_{1}\neq-1,\lambda_{2})}(v):=\begin{cases}\left(x^{*}_{1},\frac{x^{*}_{2}}{ce^{\lambda_{2}b}},\frac{x^{*}_{3}}{ce^{\lambda
b}},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right),&x^{*}_{4}x^{*}_{5}\neq 0\\\
\left(x^{*}_{1},\frac{x^{*}_{2}}{c},x^{*}_{3},0,x^{*}_{5},x^{*},y^{*}\right),&x^{*}_{4}=0\neq
x^{*}_{5}\\\
\left(x^{*}_{1},x^{*}_{2},\frac{x^{*}_{3}}{c},x^{*}_{4},0,x^{*},y^{*}\right),&x^{*}_{4}\neq
0=x^{*}_{5}\end{cases}\\\
h_{10}(v):=\begin{cases}\left(x^{*}_{1},x^{*}_{2}+\frac{d}{2x^{*}_{5}},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right),&x^{*}_{4}x^{*}_{5}\neq
0\\\
\left(x^{*}_{1},x^{*}_{2}+x^{*}_{5}\ln|x^{*}_{5}|,x^{*}_{3},0,x^{*}_{5},x^{*},y^{*}\right),&x^{*}_{4}=0\neq
x^{*}_{5}\\\
\left(x^{*}_{1},x^{*}_{2},x^{*}_{3}-x^{*}_{4}\ln|x^{*}_{4}|,x^{*}_{4},0,x^{*},y^{*}\right),&x^{*}_{4}\neq
0=x^{*}_{5}\end{cases}\\\
h_{11(\lambda)}(v):=\begin{cases}\left(x^{*}_{1},\frac{2x^{*}_{2}+x^{*}_{4}+b}{2}+\frac{\lambda
d}{2x^{*}_{4}},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right),&x^{*}_{4}x^{*}_{5}\neq
0\\\ \left(x^{*}_{1},x^{*}_{2}+\lambda
x^{*}_{5}\ln|x^{*}_{5}|,x^{*}_{3},0,x^{*}_{5},x^{*},y^{*}\right),&x^{*}_{4}=0\neq
x^{*}_{5}\\\ \left(x^{*}_{1},x^{*}_{2},x^{*}_{3}-\lambda
x^{*}_{4}\ln|x^{*}_{4}|,x^{*}_{4},0,x^{*},y^{*}\right),&x^{*}_{4}\neq
0=x^{*}_{5}\end{cases}\end{array}$

where $a=\arctan\frac{x^{*}_{4}}{x^{*}_{5}}$,
$b=\arctan\frac{x^{*}_{5}}{x^{*}_{4}}$,
$c=\left({x_{4}^{*}}^{2}+{x_{5}^{*}}^{2}\right)^{\frac{1}{1+\lambda_{1}}}$,
and
$d=\left({x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}\right)\ln\left({x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}\right)$.
We can check that $h_{8(\lambda)}$, $h_{9(\lambda_{1}\neq-1,\lambda_{2})}$,
$h_{10}$ and $h_{11(\lambda)}$ send leaves of $\mathcal{F}_{G_{13}^{0}}$ onto
those of $\mathcal{F}_{G_{13}^{\lambda}}$,
$\mathcal{F}_{G_{14}^{\lambda_{1}\lambda_{2}}}$, $\mathcal{F}_{G_{15}}$,
$\mathcal{F}_{G_{16}^{\lambda}}$, respectively. In fact, we will proof the
first case in detail, the remaining cases are absolutely similar. First, take
an arbitrary leaf $L$ of $\mathcal{F}_{G_{13}^{0}}$. Without loss of
generality, assume that $L\subset V_{3}$ as in (2.6), i.e.

$L=\left\\{v\in
V_{3}\;\Big{|}\;\frac{x^{*}_{2}x^{*}_{5}-x^{*}_{3}x^{*}_{4}}{{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}}=c;\,\,\tilde{x}^{2}_{4}+\tilde{x}^{2}_{5}\neq
0\right\\}$

where $c\in\mathbb{R}$ is some constant. For $\mathcal{F}_{G_{13}^{\lambda}}$,
consider the leaf $\tilde{L}\subset V_{3}$ as follows

$\tilde{L}=\left\\{\tilde{v}\in
V_{3}\;\Big{|}\;\frac{\tilde{x}_{2}\tilde{x}_{5}-\tilde{x}_{3}\tilde{x}_{4}}{{\tilde{x}_{4}}^{2}+{\tilde{x}_{5}}^{2}}e^{\lambda\arctan\frac{\tilde{x}_{4}}{\tilde{x}_{5}}}=c;\,\,\tilde{x}^{2}_{4}+\tilde{x}^{2}_{5}\neq
0\right\\}$

where
$\tilde{v}=(\tilde{x}_{1},\tilde{x}_{2},\tilde{x}_{3},\tilde{x}_{4},\tilde{x}_{5},\tilde{x},\tilde{y})\in
V_{3}$. By the formula of $h_{8(\lambda)}$, it is plain that
$h_{8(\lambda)}(v)=\tilde{v}$ is equivalent to

$\begin{array}[]{rl}&\tilde{v}=\begin{cases}\left(x^{*}_{1},\frac{x^{*}_{2}}{e^{\lambda
a}},\frac{x^{*}_{3}}{e^{\lambda
a}},x^{*}_{4},x^{*}_{5},x^{*},y^{*}\right),&x^{*}_{4}x^{*}_{5}\neq 0\\\
(x^{*}_{1},x^{*}_{2},x^{*}_{3},0,x^{*}_{5},x^{*},y^{*}),&x^{*}_{4}=0\neq
x^{*}_{5}\\\
(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},0,x^{*},y^{*}),&x^{*}_{4}\neq
0=x^{*}_{5}\end{cases}\\\
\Leftrightarrow&\begin{cases}\tilde{x}_{1}=x^{*}_{1},\tilde{x}_{2}=\frac{x^{*}_{2}}{e^{\lambda
a}},\tilde{x}_{3}=\frac{x^{*}_{3}}{e^{\lambda a}},\tilde{x}_{4}=x^{*}_{4}\neq
0,\tilde{x}_{5}=x^{*}_{5}\neq 0,\tilde{x}=x^{*},\tilde{y}=y^{*}\\\
\tilde{x}_{1}=x^{*}_{1},\tilde{x}_{2}=x^{*}_{2},\tilde{x}_{3}=x^{*}_{3},\tilde{x}_{4}=x^{*}_{4}=0,\tilde{x}_{5}=x^{*}_{5}\neq
0,\tilde{x}=x^{*},\tilde{y}=y^{*}\\\
\tilde{x}_{1}=x^{*}_{1},\tilde{x}_{2}=x^{*}_{2},\tilde{x}_{3}=x^{*}_{3},\tilde{x}_{4}=x^{*}_{4}\neq
0,\tilde{x}_{5}=x^{*}_{5}=0,\tilde{x}=x^{*},\tilde{y}=y^{*}.\end{cases}\end{array}$

Therefore

$\begin{array}[]{l l l l}v\in
L&\Leftrightarrow&\dfrac{x^{*}_{2}x^{*}_{5}-x^{*}_{3}x^{*}_{4}}{{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}}=c;&{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}\neq
0\\\
&\Leftrightarrow&\dfrac{\tilde{x}_{2}\tilde{x}_{5}-\tilde{x}_{3}\tilde{x}_{4}}{{\tilde{x}_{4}}^{2}+{\tilde{x}_{5}}^{2}}e^{\lambda\arctan\frac{\tilde{x}_{4}}{\tilde{x}_{5}}}=c;&\tilde{x}^{2}_{4}+\tilde{x}^{2}_{5}\neq
0\\\ &\Leftrightarrow&h_{8(\lambda)}(v)=\tilde{v}\in\tilde{L}\end{array}$

i.e. $h_{8(\lambda)}(L)=\tilde{L}$ for $L\subset V_{3}$. Therefore,
$h_{8(\lambda)}$ sends leaves of $\mathcal{F}_{G_{13}^{0}}$ onto those of
$\mathcal{F}_{G_{13}^{\lambda}}$, and all foliations
$\left(V_{3},\mathcal{F}_{G_{13}^{\lambda}}\right)$ are topological
equivalent.

To sum up, all foliations from the set in
$\big{\\{}(V_{G},\mathcal{F}_{G})/G\in\mathcal{A}_{2}\big{\\}}$ are
topologically equivalent and they determine the type $\mathcal{F}_{3}$. Thus,
the assertion in Item (1c) of Theorem 3 is proved. Moreover, the non-
equivalence of types $\mathcal{F}_{1}$, $\mathcal{F}_{2}$ and
$\mathcal{F}_{3}$ are evident since the foliated manifolds $V_{1}$, $V_{2}$
and $V_{3}$ are not homeomorphic.

  2. 2.

Now we prove the second of Theorem 3.

    1. (a)

Evidently, by [17, Theorem 23] the assertion in Item (1a) of Theorem 3 is
proved.

    2. (b)

Due to Item (1b), we only need to prove this assertion for
$\left(V_{2},\mathcal{F}_{G_{12}^{0}}\right)$.

Recall that a subset $W\subset V$ of a foliation $(V,\mathcal{F})$ is said to
be saturated (with respect to the foliation) if every leaf $L$ of
$\mathcal{F}$ has a non-empty intersection with $W$ then it lies entirely in
$W$, i.e. if $L\cap W\neq\emptyset$ then $L\subset W$. If $W\subset V$ is a
saturated submanifold then the family of all leaves $L$ of $(V,\mathcal{F})$
such that $L\subset W$ forms a new foliation on $W$ which is denoted by
$(W,\mathcal{F}_{W})$ and it is called the _restriction_ or _subfoliation_ of
$(V,\mathcal{F})$ on $W$.

By (2.4), the foliated manifold $V_{2}$ of all GMD-foliations as the disjoint
union of two open subsets ${V_{2}}_{+}$ and ${V_{2}}_{-}$ given by (2.5).
These subsets are two connected components of $V_{2}$ which are saturated with
respect to all GMD-foliations.

Now, we consider the $\left(V_{2},\mathcal{F}_{G_{12}^{0}}\right)$. For
convenience, we denote its restrictions on ${V_{2}}_{+}$ and ${V_{2}}_{-}$ by
${\mathcal{F}_{2}}_{+}$ and ${\mathcal{F}_{2}}_{-}$, respectively. Let
$p\colon V_{2}\to\mathbb{R}$ be the map that defines as follows

$p(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*}):=\left(x^{*}_{2}-\frac{x^{*}_{3}x^{*}_{4}}{x^{*}_{5}}\right)\frac{1}{e^{\frac{x^{*}_{4}}{x^{*}_{5}}}}$

where $(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in
V_{2}$. It can be verified that $p_{+}:=p|_{{V_{2}}_{+}}$ is a submersion and
$p_{+}\colon{V_{2}}_{+}\to\mathbb{R}$ is a fibration over $\mathbb{R}$ with
connected fibers. Moreover, ${\mathcal{F}_{2}}_{+}$ comes from this fibration.
Similarly, the foliation ${\mathcal{F}_{2}}_{-}$ also comes from a fibration
on $\mathbb{R}$ with connected fibers. Therefore,
$\left(V_{2},\mathcal{F}_{G_{12}^{0}}\right)$ comes from a fibration with
connected fibers on $\mathbb{R}\sqcup\mathbb{R}$, and so does the type
$\mathcal{F}_{2}$.

    3. (c)

For the assertion in Item (1c) of Theorem 3, we only need to prove for
$\left(V_{3},\mathcal{F}_{G_{13}^{0}}\right)$.

Let $p\colon V_{3}\to\mathbb{R}$ be the map that defines as follows

$p(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*}):=\frac{x^{*}_{2}x^{*}_{5}-x^{*}_{3}x^{*}_{4}}{{x^{*}_{4}}^{2}+{x^{*}_{5}}^{2}}$

where $(x^{*}_{1},x^{*}_{2},x^{*}_{3},x^{*}_{4},x^{*}_{5},x^{*},y^{*})\in
V_{3}$. It can be verified that $p$ is a submersion and $p$ is a fibration
over $\mathbb{R}$ with connected fibers. Therefore,
$\left(V_{3},\mathcal{F}_{G_{13}^{0}}\right)$ comes from a fibration with
connected fibers on $\mathbb{R}$, and so does the type $\mathcal{F}_{3}$.

The proof of Theorem 3 is complete. ∎

As an immediate consequence of Theorem 3 and Connes [1, Section 5, p. 16], we
have the following.

######  Corollary 3.3.

All Connes’ C*-algebras of GMD-foliations in Theorem 3 are determined as
follows:

  1. 1.

$C^{*}(\mathcal{F}_{1})\cong\left(C_{0}(\mathbb{R})\oplus
C_{0}(\mathbb{R})\oplus C_{0}(\mathbb{R})\oplus
C_{0}(\mathbb{R})\right)\otimes\mathcal{K}$,

  2. 2.

$C^{*}(\mathcal{F}_{2})\cong\left(C_{0}(\mathbb{R})\oplus
C_{0}(\mathbb{R})\right)\otimes\mathcal{K}$,

  3. 3.

$C^{*}(\mathcal{F}_{3})\cong C_{0}(\mathbb{R})\otimes\mathcal{K}$,

where $C_{0}(\mathbb{R})$ is the C*-algebra of continuous complex-valued
functions defined on $\mathbb{R}$ vanishing at infinity, and $\mathcal{K}$
denotes the C*-algebra of compact operators on an infinite-dimensional
separable Hilbert space.

##  4 Conclusion

We have considered connected and simply connected Lie groups corresponding to
7-dimensional Lie algebras with nilradical $\mathfrak{g}_{5,2}$. The main
results of the paper are as follows. First, we describe the picture of
maximal-dimensional K-orbits of considered Lie groups as well as their
geometric characteristics in Theorem 1, Remark 2.2 and Remark 2.4. Afterwards,
Theorem 2 proves that the families of all generic maximal-dimensional K-orbits
of considered Lie groups form measurable foliations (in the sense of Connes)
which is called GMD-foliations. Finally, the topological classification of all
GMD-foliations is given Theorem 3, and their Connes’ C*-algebras are also
described in Corollary 3.3.

## Acknowledgements

Tuyen T. M. Nguyen was funded by Vingroup Joint Stock Company and supported by
the Domestic Master/PhD Scholarship Programme of Vingroup Innovation
Foundation (VINIF), Vingroup Big Data Institute (VINBIGDATA),
VINIF.2021.TS.159. A part of this paper was done when Vu A. Le visited Vietnam
Institute for Advanced Study in Mathematics (VIASM) in summer 2022, and the
author is very grateful to VIASM for the support and hospitality.

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