Exact Kerr--Schild spacetimes from linearized kinetic gravity braiding

We generalize our recent work on k-essence sourcing Kerr--Schild spacetimes to kinetic gravity braiding scalar field. For k-essence, in order a perturbative Kerr--Schild type solution to become exact, the k-essence Lagrangian was either linear in the kinetic term (with the Kerr--Schild congruence autoparallel) or unrestricted, provided the scalar gradient along the congruence vanishes. A similar reasoning for the pure kinetic braiding contribution leads to either a vanishing Lagrangian or a scalar which is constant along the congruence. From the scalar dynamics we also derive an accompanying constraint. Finally, we discuss pp-waves, an example of Kerr--Schild spacetime generated by a constant k-essence along the Kerr--Schild congruence with vanishing Lagrangian. This allows for the construction of a Fock-type space, consisting of a tower of Kerr--Schild maps first yielding a vacuum pp-wave from flat spacetime; next a k-essence generated pp-wave from the vacuum pp-wave; and finally an arbitrary number of k-essence pp-waves with different, retarded time dependent metric functions.