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The \emph{flat deformation theorem} states that given a semi-Riemanniananalytic metric $g$ on a manifold, locally there always exists a two-form $F$,a scalar function $c$, and an arbitrarily prescribed scalar constraintdepending on the point $x$ of the manifold and on $F$ and $c$, say $\Psi (c, F,x)=0$, such that the \emph{deformed metric} $\eta = cg -\epsilon F^2$ issemi-Riemannian and flat. In this paper we first show that the above resultimplies that every (Lorentzian analytic) metric $g$ may be written in the\emph{extended Kerr-Schild form}, namely $\eta_{ab} := a g_{ab} - 2 b k_{(a}l_{b)}$ where $\eta$ is flat and $k_a, l_a$ are two null covectors such that$k_a l^a= -1$; next we show how the symmetries of $g$ are connected to those of$\eta$, more precisely; we show that if the original metric $g$ admits aConformal Killing vector (including Killing vectors and homotheties), then thedeformation may be carried out in a way such that the flat deformed metric$\eta$ `inherits' that symmetry.

Flat deformation theorem and symmetries in spacetime