  We study the topology of orbits of dynamical systems defined by
finite-dimensional representations of nilpotent Lie groups. Thus, the following
dichotomy is established: either the interior of the set of regular points is
dense in the representation space, or the complement of the set of regular
points is dense, and then the interior of that complement is either empty or
dense in the representation space. The regular points are by definition the
points whose orbits are locally compact in their relative topology. We thus
generalize some results from the recent literature on linear actions of abelian
Lie groups. As an application, we determine the generalized $ax+b$-groups whose
$C^*$-algebras are antiliminary, that is, no closed 2-sided ideal is type~I.
