  The class of 2-dimensional non-integrable flat dynamical systems has a rather
extensive literature with many deep results, but the methods developed for this
type of problems, both the traditional approach via Teichm\"{u}ller geometry
and our recent shortline-ancestor method, appear to be exclusively
plane-specific. Thus we know very little of any real significance concerning
3-dimensional systems.
  Our purpose here is to describe some very limited extensions of uniformity in
2 dimensions to uniformity in 3 dimensions. We consider a 3-manifold which is
the cartesian product of the regular octagonal surface with the unit torus.
This is a restricted system, in the sense that one of the directions is
integrable. However, this restriction also allows us to make use of a
transference theorem for arithmetic progressions established earlier by Beck,
Donders and Yang.
