In Hamiltonian dynamics and symplectic geometry a *twisted cotangent bundle* is the cotangent space $T^*N$ of a closed (compact without boundary) $n$-manifold $N$ equipped with a twisted symplectic structure: $T^*N$ carries the canonical symplectic structure $\omega=d\lambda$, where $\lambda$ is the Liouville 1-form. One can "twist" $\omega$ by adding a closed two-form $\sigma$ on $N$ as follows:
$$
\omega_{\sigma}:=\omega + \pi^*\sigma.
$$
Here $\pi:T^*N \to N$ denotes the footpoint map. It is easy to check that $\omega_{\sigma}$ is symplectic. Twisted cotangent bundles play an important role in Hamiltonian dynamics, but I am here interested in their symplectic topology. Many classical questions in symplectic topology concern the closed Lagrangian submanifolds of $(T^*N,\omega)$. But what about closed Lagrangian submanifolds in $(T^*N,\omega_{\sigma})$? Does anyone know a non-trivial example (meaning $\sigma$ is *not* exact) where $(T^*N,\omega_{\sigma})$ contains closed Lagrangian submanifolds with "good properties" (say weakly exact, monotone etc.)? Are any general statements known? Any examples, ideas, references or proofs will be highly appreciated!

It is easy to find non-compact Lagrangians in $(T^*N, \omega_{\sigma})$: If $X\subset N$ is a submanifold such that $\sigma|_{X}=0$ then its conormal space $$ \nu^*(X):=\{ p\in T^*N\ |\ p|_{TX}\equiv 0 \}\subset T^*N $$ is a non-compact Lagrangian submanifold. My questions therefore concerns closed Lagrangian submanifolds! It is easy to find closed Lagrangians when $\sigma$ is exact. Hence, my interest is really in the case when $\sigma$ is not exact.

Thanks in advance!