We study connected orientable spacelike hypersurfaces $x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ where $L_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a fixed integer $0leq k

It is shown that the only vacuum algebraically special space-time that is asymptotically simple is Minkowski space. The structure of null infinity for algebraically special space-times satisfying a subset of the vacuum equations is shown to be particularly simple in terms of the coordinates used in Kerr's original reduction of the field equations. With the assumption of the existence of a globa...