I do not quite understand the idea of an integrable dynamical system. Does it mean that the EOMs are analytically and exactly solvable? What are the necessary and sufficient conditions such that a system is integrable?
For mathematicians the question is very hard, for physicists the following jargon is used to intuitively define integrability: solvable=integrable, separable=integrable, existence of a Lax pair=integrable, # of conserved quantities equats to # of degrees of freedom = integrable.
It is interesting that the mathematicians have managed to find the exceptions to all the intuitive definitions. In general the most general physical definition is the following: if there is no of chaos then the system is integrable.
For more information look at Zakharov V.E. What is integrability (Springer, 1992).