If $X$ is a symplectic space and $H$ is a Hamiltonian on $X$, then we have the non-normalized Gibbs measure $e^{-\lambda H}dm$ for any $\lambda\in\mathbb R$ with $dm$ being the Haar measure on $X$, which is a family of invariant measures for the Hamiltonian flow.
I am seeking examples of invariant measures for various Hamiltonian partial differential equations (PDE's). If you name a certain PDE and a family of invariant measures for it, please explain whether we know if those are the only invariant measures possible for that PDE.
