Integrable asymmetric $\lambda$-deformations

We construct integrable deformations of the $\lambda$-type for asymmetrically gauged WZW models. This is achieved by a modification of the Sfetsos gauging procedure to account for a possible automorphism that is allowed in $G/G$ models. We verify classical integrability, derive the one-loop beta function for the deformation parameter and give the construction of integrable D-brane configurations in these models. As an application, we detail the case of the $\lambda$-deformation of the cigar geometry corresponding to the axial gauged $SL(2,R)/U(1)$ theory at large $k$. Here we also exhibit a range of both A-type and B-type integrability preserving D-brane configurations.