From conformal Killing vector fields to boost-rotational symmetry

We discuss a connection between three-dimensional Riemannian manifolds (Sigma, gamma) admitting a special conformal Killing vector field and static vacuum or non-vacuum spacetimes. Any such (Sigma, gamma) generates a vacuum spacetime (M, g) but it also generates a spacetime (M, g, Phi), where (g, Phi) satisfies the Einstein-Klein-Gordon massless minimally coupled gravity equations, or the Einstein-Conformal scalar field equations. The resulting spacetimes either admit four Killing vector fields or possess boost and rotational symmetry. We argue that this connection goes beyond the vacuum or Einstein-scalar field system and it should be viewed as a mechanism of generating solutions for the Einstein equations, admitting a hypersurface orthogonal Killing vector field.