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 $K_i$ such that $K_1=H$ and $[K_i,K_j]=0$ for all $i,j=1,\dots,n$. 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 $K_j$ for all $j=1,\dots,n$.
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