  In the Maxwell interaction model the collision rate is independent of the
relative velocity of the colliding pair and, as a consequence, the collisional
moments are bilinear combinations of velocity moments of the same or lower
order. In general, however, the drift term of the Boltzmann equation couples
moments of a given order to moments of a higher order, thus preventing the
solvability of the moment hierarchy, unless approximate closures are
introduced. On the other hand, there exist a number of states where the moment
hierarchy can be recursively solved, the solution generally exposing
non-Newtonian properties. The aim of this paper is to present an overview of
results pertaining to some of those states, namely the planar Fourier flow
(without and with a constant gravity field), the planar Couette flow, the
force-driven Poiseuille flow, and the uniform shear flow.
