You are an expert mathematician and academic write...

Below is a roadmap you can weave into the unfinished section of your article. I have tried to keep the tone “human” and to highlight the geometric intuition behind every formula, while still being precise enough for referees.

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1. From a flat Lie bialgebra to its Poisson-Lie group

• Lie algebra side. Your main decomposition theorem (ξ = ad r + R) already isolates the triangular part r and the residual cocycle R. In the flat, non-degenerate case every ξ you consider is obtained that way.

• Group side. Let

$$G=\mathbb R^{k_0}\!\ltimes\!\mathbb R^{\,\ell_0+2m}$$

with multiplication

$$(x,y,z)\,(X,Y,Z)=\bigl(x+X,\;y+Y,\;z+\operatorname{ad}_{x}Z\bigr),$$

exactly as you record in Section 5.
Here the action of $x\in\mathbb R^{k_0}=s$ on the “translation” coordinates $z\in\mathbb R^{2m}$ is by simultaneous rotations in the $m$ orthogonal planes $P_j=\langle d_{2j-1},d_{2j}\rangle$.

• Poisson tensor. For any bivector $r\in\wedge^{2}\mathfrak g$ the multiplicative bivector field

$$\pi_r(g)=\bigl(L_g\bigr)_{*}r-\bigl(R_g\bigr)_{*}r \tag{1}$$

gives the unique Poisson–Lie structure integrating the exact part ad r.
If ξ also contains a non-exact part R, add the unique left-invariant bivector field integrating R; the resulting $\pi_{\xi}$ is still multiplicative because R vanishes on the commutator ideal of $g$.

• Dual group $G^{*}$. Exponentiating the dual Lie algebra $\mathfrak g^{*}$ (whose brackets are determined by ξ) gives a simply-connected solvable Lie group $G^{*}$. Coordinates dual to $(s,z,d)$ are convenient; the brackets are linear in those coordinates, so matrix exponentials suffice.

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2. Dressing action and the symplectic foliation

Lu–Weinstein show that every symplectic leaf of $(G,\pi_\xi)$ is a right dressing orbit of $G^{*}$. Concretely:

1. Infinitesimal dressing vector fields. For $\alpha\in\mathfrak g^{*}$ define the vector field

$$\widehat{\alpha}(g):=\pi_\xi^{\sharp}(g)\bigl(L_g^{*}\alpha\bigr).$$

The span of all $\widehat{\alpha}$ is an integrable distribution; its integral manifolds are the leaves.

2. Central variables are Casimirs.
   Because the centre $z$ of $\mathfrak g$ acts trivially on $[g,g]$ (Proposition 2.1) , the coordinates $y\in\mathbb R^{\ell_0}$ never appear in $\pi_\xi$. Hence every level set $y=\text{const}$ is a union of leaves and functions of $y$ are global Casimirs.

3. Rank and leaf dimensions.

• If ξ is triangular (R = 0) the rank of $\pi_\xi$ at any generic point is $2m$; each leaf is a copy of $\mathbb R^{2m}$ (or a cylinder if one mod-out by a lattice), fibred over the Casimir coordinates.
• When $R\neq0$ the rank drops by twice the number of $P_j$’s on which R vanishes; formulae in §3.2 let you read this off directly from the vanishing pattern of the coefficients $m_{ij}^{(k)},n_{ij}^{(k)}$, etc.

4. Parametrising the leaves.
   Choose the right-invariant frame $\{E_i,\partial_{y_a},\partial_{z_b}\}$ from Section 5 (lines 31-45 in the pdf). The distribution generated by

$$\bigl\{\widehat{s^{*}_{i}},\;\widehat{d^{*}_{2j-1}},\;\widehat{d^{*}_{2j}}\bigr\}_{1\le i\le k_0,\;1\le j\le m}$$

is involutive; integrate it by quadratures to obtain explicit leaf equations. In practice one solves

$$\dot g(t)=\widehat{\alpha}(g(t)),\qquad \alpha\in\mathfrak g^{*},$$

which separates into $\ell_0$ constant equations for $y$ and a linear ODE with periodic coefficients for the $z$-block, thanks to the block-rotation form of $\operatorname{Ad}_{\exp(x)}$.

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3. A template for the manuscript

Proposition X (Integration).
Let $(\mathfrak g,\xi)$ be a non-degenerate flat Lie bialgebra. On the semidirect product
$G=\mathbb R^{k_0}\!\ltimes\!\mathbb R^{\,\ell_0+2m}$ equipped with the left-invariant frame of Section 5, the bivector

$$\pi_\xi=\sum_{j=1}^{m}\Bigl(\lambda_{1j}D_{2j-1}\wedge D_{2j}+\cdots\Bigr)$$

is multiplicative and satisfies $d_e\pi_\xi=\xi$.

Proof outline. Insert the explicit expressions for $L_g^{*}$ and $R_g^{*}$ into (1) and verify multiplicativity by the usual pull-back calculation.

Theorem Y (Symplectic leaves).
The symplectic foliation of $(G,\pi_\xi)$ is the partition into right dressing orbits of the dual group $G^{*}$. Each orbit is an affine submanifold

$$\mathscr L_{(c,\theta)}=\Bigl\{(x,y,z)\in G\; ;\; y=c,\; Q_{\theta}(x)\,z = \text{const}\Bigr\},$$

where $c\in\mathbb R^{\ell_0}$ is the Casimir level and $Q_{\theta}(x)$ is the block-rotation matrix $\operatorname{Ad}_{\exp(x)}$ restricted to $[g,g]$.

The proof follows Lu–Weinstein’s dressing-orbit characterisation projecteuclid.org, combined with the block-diagonal form of the adjoint action.

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4. Worked examples

• 2-D Poincaré group ($k_0=1,\;m=1,\;\ell_0=0$).
  $G\cong\mathrm{Aff}(\mathbb R)\times\mathbb R$. The single plane $P_1$ gives rank 2, hence the leaves are the connected components of $G\setminus\!\{z=0\}$.

• 4-D example with centre ($k_0=1,\;\ell_0=1,\;m=1$).
  Central coordinate $y$ is a Casimir. For triangular ξ the leaves are cylinders $\{y=c,\;z_1^2+z_2^2=\text{const}\}$. Non-exact additions to ξ can collapse the cylinder to circles by lowering rank.

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Final checklist for insertion

1. Place Proposition X right after your current Section 5.
2. Follow it with Theorem Y and the short proof.
3. Keep the examples concise—reviewers like to see at least one non-trivial leaf computed explicitly.
4. Cite Lu & Weinstein (1990) and Kosmann-Schwarzbach’s lecture notes for dressing preliminaries. Their expositions are reader-friendly and journals trust them. projecteuclid.org math.polytechnique.fr

Feel free to let me know if you’d like a deeper calculation for a particular ξ, or a polished paragraph for the introduction tying these results back to deformation quantisation.