If we deal with quantum finite-dimensional systems without spin, the definition is this: if we have a system with n degrees of freedom whose (quantum) Hamiltonian is given by an operator H, then this system is called integrable if there exist n independent operators Ki such that K1=H and [Ki,Kj]=0 for all i,j=1,…,n. All operators, of course, are assumed to be (formally) self-adjoint.
The matter of how one should interpret the word "independent" here is a bit tricky. Linear independence is not sufficient, and we should at least require the functional independence of classical limits of Kj for all j=1,…,n.
For further details see e.g. Definition 6 in this paper by Miller, Post and Winternitz and references therein.
This post imported from StackExchange Physics at 2014-04-25 13:36 (UCT), posted by SE-user just-learning