Flat deformation theorem and symmetries in spacetime

The \emph{flat deformation theorem} states that given a semi-Riemannian analytic metric $g$ on a manifold, locally there always exists a two-form $F$, a scalar function $c$, and an arbitrarily prescribed scalar constraint depending on the point $x$ of the manifold and on $F$ and $c$, say $\Psi (c, F, x)=0$,suchthatthe\mathit{\text{deformed}}\mathit{\text{metric}}$\eta = cg -\epsilon F^2$ is semi-Riemannian and flat. In this paper we first show that the above result implies 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$\eta$isflatand$k_a, l_a$aretwonullcovectorssuchthat$k_a l^a= -1$;nextweshowhowthesymmetriesof$g$areconnectedtothoseof$\eta$,moreprecisely;weshowthatiftheoriginalmetric$g$ admits a Conformal Killing vector (including Killing vectors and homotheties), then the deformation may be carried out in a way such that the flat deformed metric $\eta$ `inherits’ that symmetry.