It is a well known fact that Type IIB SUGRA gives a solution $AdS_5 \times S^5$ supported by the RR five-form flux whose integral gives $\int_{S^5}F_5 = (4\pi \alpha')^2N$. I would like to know exactly what fileds support a "cousin" geometry, namely the $AdS_3 \times S^3 \times S^3 \times S^1 $ and $AdS_3 \times S^3 \times S^3 \times \mathbb{R} $. It is known that such a background is created by two D1-D5 brane systems (see for example Tong 1402.5135). I think that each of the $S^3$ should be supported by an $F_3$ (or $ H_3$) as for $AdS_3$ but I am not really sure about the rest. What other background fields can be there? Finally, if I plug them into the EOM they should satisfy everything just like the case of $AdS_5 \times S^5$, right?

Any help would be welcome!