Not understanding the foliation structure of a Poisson manifold

Question

Consider a Poisson manifold $(M, P)$ (no assumption about the rank or regularity of $P$). For each possible rank $2r$ of $P$, let $(M_r, P_r)$ be the corresponding symplectic leaf of dimension $2r$, with symplectic form $\Omega_r$ and associated non-degenerate Poisson bivector $P_r$.

1. What is (if any) the relationship between $P_r$ and $P$? Is $i_r$ a Poisson map?

2. Is $P_r$ unique (modulo symplectic isomorphisms of $M_r$)? I still am not able to follow the proof of its existence, but it (in fact its associated symplectic form) seems to be obtained through a glueing process, and this approach does not always produce unique results.

3. If $x \in M_r$, then the Darboux-Weinstein theorem essentially says that there exist coordinates $(x_1, \dots, x_r, p_1, \dots, q_r, z_1, \dots, z_s)$ around $x$ such that in these coordinates

   $$P = \sum \limits _{i = 1} ^r \frac {\partial} {\partial x_i} \wedge \frac {\partial} {\partial p_i} + \sum \limits _{i = 1} ^s \varphi _{ij} \frac {\partial} {\partial z_i} \wedge \frac {\partial} {\partial z_j}$$

   and $\varphi _{ij} (x) = 0$. If $P_x$ is the first summand, what is the relationship between $P_x$, $P_r$ and $P$? (Alternatively, if $\Omega_x$ is the symplectic form associated to $P_x$, what is the relationship between $\Omega_x$ and $\Omega_r$)?

4. Are the coordinates $(x_1, \dots, x_r, p_1, \dots, q_r)$ introduced above Darboux coordinates around $x$ when seen as a point of $M_r$?

Answer 1

To 1. & 2.: Yes, the map $i_r $ is a Poisson map so we have $(i_r)_* P_r = P\circ i_r $. Moreover $i_r $ is an immersion and hence $P_r$ is unique.

To 3.: If $P_x$ is the first summand then you have $P_x \circ i_r = (i_r)_* P_r $. An important remark is that the theorem actually states that $\phi_{ij}=\phi_{ij}(z)$ with $\phi_{ij}(0)=0$. So locally around a point $x\in M$ you can find a transversal $x\in T$ to the symplectic leaf $x\in M_r$ such that $P$ splits into a Poisson structure on an open neighborhood of $x $ in $M_r$, given here by the first summand, and a Poisson structure on an open around $x\in T$, given by the second summand, such that $x $ is a symplectic leaf of the corresponding foliation on this open in T.

To 4.: Yes, which should follow from the explainations above.