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The most concrete version of the question is :

A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. Is it true that every orbit on $L$ is periodic of period $T$?

How about if we also assume that $L$ is a graph?

A more fanciful version of the question is whether there are any obstructions to a flow on a torus $L$ to be realized as the restriction of a Hamiltonian flow to an invariant Lagrangian torus.

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