An Liouville integrable system admits a set of action-angle variables and is by definition non-chaotic. Is the converse true however, are non-integrable systems automatically chaotic? Are there any examples of non-integrable systems that are not chaotic?
The key point here is that, any dynamical system that is not completely integrable will exhibit chaotic regimes1. In other words not all orbits will lie on an invariant torus (Liouville's torus is the topological structure of a fully integrable system), in principle a chaotic system can even have closed stable periodic orbits (typical for regular/integrable systems) for some initial conditions, the set of such conditions has measure zero (meaning the states on that orbit are only reachable from other states of the same orbit).
In order to get yourself acquainted with such concepts, I suggest looking into 2D dynamic billiards. These models are of great interest because their dynamics are solely defined by the shape of the boundary, circular, ellipsoid, stadium etc. Now an interesting example to showcase here would be the oval shaped boundary (note circular and ellipsoid billiards are regular because of their symmetries):
In the above image (by Tureci, Hakan, et al. 2002), on left you see the poincaré map2 of the 2D oval billiard (with specular reflection), and on the right you see 3 examples of different regimes of the system. This is a perfect example showcasing a system that admits only locally integrable regions. Case a) corresponds to a quasi-periodic orbit, only marginally stable. Case b) shows a stable periodic orbit surrounded by a stable island and finally case c) corresponding to the entirety of densely dotted regions of the map, is indicative of chaotic motion. For further reading, I suggest looking into some of the articles on scholarpedia, and of course not to miss this fantastic review by A. Douglas Stone.
1For example all non-linear systems that are not Liouville integrable (as explained in comments). Note that linear systems can always be solved by exponentiation. But that said one must be wary of distinctions between solvability and integrability.
2These maps are obtained by choosing a poincaré section, and finding the intersection of trajectories in phase space with this section. Such maps allow for a representation of the evolution of any dynamical system, regardless of the dynamics involved. For more intuition, see here.