I'm a PhD student in maths and I am intersted in classical integrable systems (integrable hierarchies). I am currently preparing an introduction to classical integrable systems and I was asking myself the following question.
Are there simple non-integrable systems (say in the sense of Liouville: complete integrability) in which we can see the consequences.
To be more precise, I am searching for a system of PDEs that does not have sufficiently many hamiltonians and for which we cannot find the exact solutions (as a consequence).
I believe we really understand a theorem only once we know a counterexample and see how wrong it goes without the proper hypothesis. That's why I am looking for that kind of systems.
Most probably my problem is not very well defined so we could first try to ask a better question ;)
Have good day, Clément Justin ${}$