Kerr-Schild perturbations of coset CFTs as scale invariant integrable $\sigma$-models

Kerr-Schild perturbations in General Relativity provide a fruitful way of constructing new exact solutions starting from known ones, elucidating also the structure of the spacetimes. We initiate such a study in the context of string theory and supergravity. Specifically, we explicitly construct Kerr-Schild perturbations of coset CFTs based on low dimensionality orthogonal groups. We show that these give rise to scale, but not Weyl, invariant integrable $\sigma$-models. We explicitly demonstrate that these models can also be derived from a particular limiting procedure of $\lambda$-deformed coset CFTs based on non-compact groups. The target space of the simplest $\sigma$-model describes a two-dimensional scale invariant black hole for which we also provide two different embeddings to type-II supergravity.