K-essence sources of Kerr-Schild spacetimes

We extend a result by one of the authors, established for nonvacuum Einstein gravity, to minimally coupled k-essence scalar-tensor theories. First we prove that in order to source a Kerr-Schild type spacetime, the k-essence Lagrangian should be at most quadratic in the kinetic term. This is reduced to linear dependence when the Kerr-Schild null congruence is autoparallel. Finally, we show that requiring the solutions of the Einstein equations linearized in Kerr-Schild type perturbations to also solve the full nonlinear system of Einstein equations, selects once again k-essence scalar fields with Lagrangians linear in the kinetic term. The only other k-essence sharing the property of sourcing perturbative Kerr-Schild spacetimes which are also exact, is the scalar field constant along the integral curves of the Kerr-Schild congruence, with otherwise unrestricted Lagrangian.