  We prove that every homogeneous flow on a finite-volume homogeneous manifold
has countably many independent invariant distributions unless it is conjugate
to a linear flow on a torus. We also prove that the same conclusion holds for
every affine transformation of a homogenous space which is not conjugate to a
toral translation. As a part of the proof, we have that any smooth partially
hyperbolic flow on any compact manifold has countably many distinct minimal
sets, hence countably many distinct ergodic probability measures. As a
consequence, the Katok and Greenfield-Wallach conjectures hold in all of the
above cases.
