  The aim of this paper is to study cohomogeneity one isometric linear actions
on the $p+q$-dimensional pseudo-Euclidean space $\mathbb{R}^{p,q}$. It is
proved that the natural isometric action of the nilpotent factor of an Iwasawa
decomposition of $SO(p,q)$ is not of cohomogeneity one. The orbits of
cohomogeneity one actions of some subgroups of a maximal parabolic subgroup of
the isometry group of $\mathbb{R}^{p,q}$ are determined and it is proved that
there exist cohomogeneity one isometric actions on $\mathbb{R}^{p,q}$ which are
orbit-equivalent on the complement of a $p$-dimensional degenerate subspace
$\mathbb{W}^p$ of $\mathbb{R}^{p,q}$ and not orbit-equivalent on $\mathbb{W}p$.
