Assume that $n$ is an even number. What is a natural symplectic structure on $M_{n}(\mathbb{R})$, such that for every $1\leq k \leq n$ the manifold of $k$-rank matrices would be invariant under the flow of the Hamiltonian vector field correspond to the Hamiltonian "$H=$ Determinant"? (Or at least the manifold of $n-1$ rank matrices would be invariant under this Hamiltonian vector field)

We consider a contact analogy as follows:

Assume that $n$ is an odd number. Under what contact structures on $M_{n}(\mathbb{R})$, the set of singular matrices is invariant under the flow of corresponding Reeb vector field? Or a refined version: the set of $k$ rank matrices would be invariant under this flow.