Flat deformation of a spacetime admitting two commuting Killing fields

It is shown that, given an analytic Lorentzian metric on a 4-manifold, gab, which admits two Killing vector fields, there exists a local deformation law ηab = a gab + b Hab, where Hab is a two-dimensional projector, such that ηab is flat and admits the same Killing vectors. We also characterize the particular case when the projector Hab coincides with the quotient metric. We apply some of our results to general stationary axisymmetric spacetimes.