  It is well known that Hopf-fibre T-duality and uplift takes the D1-D5
near-horizon into a class of $AdS_3 \times S^2$ geometries in 11D where the
internal space is a Calabi-Yau three-fold. Moreover, supersymmetry dictates
that Calabi-Yau is the only permissible $SU(3)$-structure manifold.
Generalising this duality chain to non-Abelian isometries, a strong parallel
exists, resulting in the first explicit example of a class of $AdS_3 \times
S^2$ geometries with $SU(2)$-structure. Furthermore, the non-Abelian T-dual of
$AdS_3 \times S^3 \times S^3 \times S^1$ results in a new supersymmetric $AdS_3
\times S^2$ geometry, which falls outside of all known classifications. We
explore the basic properties of the holographic duals associated to the new
backgrounds. We compute the central charges and show that they are compatible
with a large $\mathcal{N}=4$ superconformal algebra in the infra-red.
