  We investigate classical integrability on two recently discovered classes of
backgrounds in massive IIA supergravity. These vacua are of the form
AdS$_3\times\,$S$^2\times\mathbb{R}\times\,$CY$_2$, they preserve small
$\mathcal{N}=(0,4)$ supersymmetry and are associated with D8$-$D6$-$D4$-$D2
Hanany-Witten brane set-ups. We choose an appropriate string embedding and use
differential Galois theory on its associated Hamiltonian system, intending to
produce the conditions under which Liouvillian solutions may occur. By
constraining the parameters of the system according to the consistency of the
associate brane set-ups we prove that no such conditions exist, yielding the
complete non-integrability of these vacua. That is, up to the trivial cases
where the background reduces to the Abelian and non-Abelian T-dual of
AdS$_3\times\,$S$^3\times\,$T$^4$.
