All integrable Hamiltonian systems are alike, while each nonintegrable one is nonintegrable in its own way
cropping
The vfplot program plots arrows up to the boundary, treating that boundary as a wall against which the arrows should be packed carefully. But for some vector fields the boundary is just a selection from a larger theoretical model, possibly infinite or semi-infinite. This extent "beyond the plot" can be indicated visually by cropping the plot on one-or-more sides.
The Boussinesq flow plot crops just the top and bottom of the field, indicating and pipe which is vertically unbounded.