  We describe the global dynamics of some pointwise periodic piecewise linear
maps in the plane that exhibit interesting dynamic features. For each of these
maps we find a first integral. For these integrals the set of values are
discrete, thus quantized. Furthermore, the level sets are bounded sets whose
interior is formed by a finite number of open tiles of certain regular or
uniform tessellations. The action of the maps on each invariant set of tiles is
described geometrically.
