I am interested in generalizing some aspects of the ergodic hierarchy (of classical dynamical systems) to quantum theory. However, while I understand the definitions of the different levels of the hierarchy: $$ \mbox{Bernoulli} \subset K-\mbox{ergodic} \subset \mbox{Strongly mixing} \subset \mbox{Weakly mixing}\subset \mbox{Merely ergodic} $$ and know plenty of examples of K-ergodic (e.g., Bunimovich stadium) and strongly mixing (e.g., irrational trianglular billiard) systems, I have not actually seen any reasonable examples of the weakly mixing and merely ergodic levels. By "reasonable," I mean something like a billiard or a Hamiltonian dynamical system (for my purposes, I need something, which would be straightforward enough to quantize).

My question is: is somebody familiar with concrete examples of classical dynamical systems representing weakly mixing and merely ergodic levels?