In this paper we address the problem of describing in explicit algebraic terms the collective structure of the coadjoint orbits of a connected, simply connected exponential solvable Lie group
G
G
. We construct a partition
℘
\wp
of the dual
g
∗
{\mathfrak {g}^{\ast } }
of the Lie algebra
g
\mathfrak {g}
of
G
G
into finitely many
Ad
∗
(
G
)
\operatorname {Ad}^{\ast } (G)
-invariant algebraic sets with the following properties. For each
Ω
∈
℘
\Omega \in \wp
, there is a subset
Σ
\Sigma
of
Ω
\Omega
which is a cross-section for the coadjoint orbits in
Ω
\Omega
and such that the natural mapping
Ω
/
Ad
∗
(
G
)
→
Σ
\Omega /\operatorname {Ad}^{\ast } (G) \to \Sigma
is bicontinuous. Each
Σ
\Sigma
is the image of an analytic
Ad
∗
(
G
)
\operatorname {Ad}^{\ast }(G)
-invariant function
P
P
on
Ω
\Omega
and is an algebraic subset of
g
∗
{\mathfrak {g}^{\ast }}
. The partition
℘
\wp
has a total ordering such that for each
Ω
∈
℘
\Omega \in \wp
,
∪
{
Ω
′
:
Ω
′
≤
Ω
}
\cup \{ \Omega \prime :\Omega \prime \leq \Omega \}
is Zariski open. For each
Ω
\Omega
there is a cone
W
⊂
g
∗
W \subset {\mathfrak {g}^{\ast } }
, such that
Ω
\Omega
is naturally a fiber bundle over
Σ
\Sigma
with fiber
W
W
and projection
P
P
. There is a covering of
Σ
\Sigma
by finitely many Zariski open subsets
O
O
such that in each
O
O
, there is an explicit local trivialization
Θ
:
P
−
1
(
O
)
→
W
×
O
\Theta :{P^{ - 1}}(O) \to W \times O
. Finally, we show that if
Ω
\Omega
is the minimal element of
℘
\wp
(containing the generic orbits), then its cross-section
Σ
\Sigma
is a differentiable submanifold of
g
∗
{\mathfrak {g}^{\ast } }
. It follows that there is a dense open subset
U
U
of
G
∅
^
G\hat \emptyset
such that
U
U
has the structure of a differentiable manifold and
G
∅
^
∼
U
G\widehat \emptyset \sim U
has Plancherel measure zero.