  It is shown that given an analytic Lorentzian metric on a 4-manifold, $g$,
which admits two Killing vector fields, then it exists a local deformation law
$\eta = a g + b H$, where $H$ is a 2-dimensional projector, such that $\eta$ is
flat and admits the same Killing vectors. We also characterize the particular
case when the projector $H$ coincides with the quotient metric. We apply some
of our results to general stationary axisymmetric spacetimes
