  We study properly discontinuous and cocompact actions of a discrete subgroup
$\Gamma$ of an algebraic group $G$ on a contractible algebraic manifold $X$. We
suppose that this action comes from an algebraic action of $G$ on $X$ such that
a maximal reductive subgroup of $G$ fixes a point. When the real rank of any
simple subgroup of $G$ is at most one or the dimension of $X$ is at most three,
we show that $\Gamma$ is virtually polycyclic. When $\Gamma$ is virtually
polycyclic, we show that $\Gamma$ is virtually polycyclic. When $\Gamma$ is
virtually polycyclic, we show that the action reduces to a NIL-affine
crystallographic action. As applications, we prove that the generalized
Auslander conjecture for NIL-affine actions holds up to dimension six and give
a new proof of the fact that every virtually polycyclic group admits a
NIL-affine crystallographic action.
