Non-trivial outer algebra automorphisms may be utilized in λ-deformations of (gauged) WZW models thus providing an efficient way to construct new integrable models. We provide two such integrable deformations of the exact coset CFT $SU{\left(2\right)}_k\times U\left(1\right)/U{\left(1\right)}_q$ with a vector and axial residual gauge. Besides the integer level k and the deformation parameter λ, these models are characterized by the embedding parameter q of the $U\left(1\right)$ factor. We show that an axial-vector T-duality persists along the deformations and, therefore, the models are canonically equivalent. We demonstrate integrability even though the space is non-symmetric and compute the RG-flow equations for the parameters λ and q. Our example provides an integrable deformation of the gravitational solution representing a Euclidean three-dimensional black string.