Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$. We denote by $\mathfrak{g}^*$ the dual of $\mathfrak{g}$. Let $(M, \omega)$ be a symplectic manifold with a Hamiltonian $G$-action. Let $\mu : M \rightarrow \mathfrak{g}^*$ be the associated equivariant moment map. For $\lambda \in \mathfrak{g}^*$ we write $G.\lambda$ for the coadjoint orbit.
Theorem (Guillemin, Sternberg). Let $\lambda \in \mathfrak{g}^*$ and $Z \subset \mathfrak{g}^*$ a submanifold perfectly transverse to $G.\lambda$ at $\lambda$, that is, such that $T_\lambda Z \oplus T_\lambda (G.\lambda) = \mathfrak{g}^*$. Then, if $p \in M$ with $\mu(p) = \lambda$, there is a neighborhood $U$ of $p$ such that $\mu^{-1}(Z) \cap U$ is a symplectic submanifold of$M$.
Why does the submanifold $Z$ exist ? Does always exists a submanifold of a manifold $M$ which is transverse to another submanifold of $M$ ?