  We consider warped AdS$_3\times $S$^2\times$M$_5$ backgrounds in type II
supergravity preserving small ${\cal N}=(4,0)$ supersymmetry. We show that
imposing ${\cal N}=(4,0)$ supersymmetry imposes between 0 and 3 a priori
isometries in the internal M$_5$. In this work we focus on classes of solution
where M$_5$ exhibits no a prioi isometry which imposes additional constraints.
Solving these in IIA forces M$_5$ to support an SU(2)-structure, a case already
studied in arXiv:1908.09851, while in IIB one arrives at two broad new classes
with identity structure that we reduce to local expressions for the physical
fields and PDEs.
