Consider Arnold's example for Arnold diffusion 1964. $$H=I_1^2/2+I_2^2/2+\epsilon(1-\cos\theta_2)(1+\mu(\sin\theta_1+\sin t)) $$

We can first make it a system of three degrees of freedom.

Then we know this system is not integrable in the Liouville Arnold sense. One integral is the Hamiltonian. Can we find one more integral?

I think the answer should be no. But it is not easy to prove. Maybe one more piece of information is useful. In Arnold's 1964 paper, he calculated two different nonvanishing Melnikov functions.