In my problem I have non autonomous Hamiltonian which depends on 2 parameters (pretty close to oscillator Hamiltonian, $(a+b\cos t +1) p^2+(a+b\cos t1)q^2$, $a,b$  parameters). From numerical experiments and by some other ways its proven that for some values of parameters the monodromy matrix $M$ is identity  all solutions are periodic with period $2\pi$. Is there any kind of necessary condition for Hamiltonian so this could happen? For example its clear from numerical experiments that this happens only for integer values of $a$ what is pretty strange.
