  We discuss marginal deformations of warped AdS$_3\times$S$^2$ solutions
preserving small $\mathcal{N}=(0,4)$ supersymmetry in massive IIA and
eleven-dimensional supergravity and obtain a whole family of new solutions. We
characterise these new backgrounds by studying some observables like the
quantised charges, associated Hannany-Witten brane set-ups and the holographic
central charge, the latter is shown to be invariant under the deformation. The
study of the preservation of supersymmetry shows that the new backgrounds
support an identity structure on the internal five-dimensional space, which is
dynamical.
