Let $D$ be a differential operator on the space of smooth functions on a manifold $M$. The symbol of $D$ can be considered as a Hamiltonian on the cotangent bundle $T^*M$. We call this Hamiltonian as "Corresponding symbol Hamiltonian"

Motivated by the above interesting linked question and this post and this one we ask the following question:

Is there an elliptic operator on a manifold whose corresponding symbol Hamiltonian has an isolated periodic orbit?

Note: We add the ellipticity condition since we learn from this answer that for differential operator associated with a vector field, which is a non elliptic operator, we do not have an isolated periodic orbit