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

On Cohomogeneity One Linear Actions On Pseudo-euclidean Space $\mathbb{R}^{p,q}$