AdS$_3\times$ S$^2$ in IIB with small ${\cal N}=(4,0)$ supersymmetry

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.