classical mechanics - Are there necessary and sufficient conditions for ergodicity?

What are the necessary and sufficient conditions (if any) for ergodicity (or non-ergodicity)?

I see for instance that some integrable systems are not ergodic. For instance a linear chain of harmonic oscillators put to oscillate in a determinate normal mode will remain in that mode. If I add non-linearity it breaks down integrability and (perhaps) the system becomes ergodic. Thus I suspect that concepts such as integrability, chaos and non-linearity are intimately related to ergodicity. Is there any set of conditions relating these concepts with ergodicity?