I would refer you to the version for hamiltonian systems of a result known as Poincaré–Lyapunov theorem that describes the periodic orbits around a known one when a certain condition is satisfied.

> Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold, an $H$ a smooth regular function on $M$.
> 
> Let $\Lambda$ be a $1$-dimensional compact connected submanifold of $M$ which is invariant under the flow of $X_H$, i.e. $\Lambda$ is the image of a periodic integral curve of $X_H$.
> 
> If $1$ is not an eigenvalue for the derivative of the first recurrence map for $X_H$ in a point of $\Lambda$ then there exists a $2$-dimensional symplectic submanifold $N$ of $(M,\omega)$ containing $\Lambda$ such that $H|_N$ is a summersion whose fibers are compact connected and invariant under the flow of $X_H$.

So under the stated non-degeneracy condition a periodic trajectory of $X_H$ is included in a family of periodic orbits forming a symplectic submanifold and parametrized by $H$.

For a reference and a generalization which joins together the Poincaré–Lyapunov theorem with the Liouville–Arnol'd theorem, I would suggest N.N. Nekhoroshev: [The Poincaré–Lyapunov–Liouville–Arnol'd theorem][1]. Funct. Anal. Appl. 28 (1994), no. 2, 128–129. [Zbl 0847.58035][2], [MR1283258][3].


  [1]: https://doi.org/10.1007/BF01076504
  [2]: https://zbmath.org/?q=an:0847.58035
  [3]: https://mathscinet.ams.org/mathscinet-getitem?mr=1283258