Let $\Gamma$ be a subgroup of the group of affine transformations of the affine space ${\mathbb R}^{2n+1}$. Suppose $\Gamma$ acts properly discontinuously on ${\mathbb R}^{2n+1}$. The paper deals with the question which subgroups of $\mathrm{GL}(2n+1,{\mathbb R})$ occur as Zariski closure $\ov{\ell(\Gamma)}$ of the linear part of such a group $\Gamma$. The two main results of the paper say that $\mathrm{SO}(n+1,n)$ does occur as $\ov{\ell(\Gamma)}$ of such a group $\Gamma$ if $n$ is odd, but does not if $n$ is even.