  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.
