SUGRA in $AdS_3 \times S^3 \times S^3 \times R$

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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!

As far as I can see, there are only three kinds of sources: $Q_1$ D1 branes and two types of D5 branes, $Q_5^\pm$. Naively, one expects the metric, dilaton to be non-zero. You will need to do some analysis of the condition on the unbroken supersymmetry to check if you need to turn on the B-field as well.
Recall how one got $AdS_5\times S^5$ from the sugra solution for N D3-branes. As Maldacena showed, one needs to take a near horizon limit of the sugra solution to get to $AdS_5\times S^5$. I suspect that you need to do the same -- first find the supergravity solution for your configuration of D-branes and then take the near horizon limit to obtain $AdS_3\times S^3\times S^3 \times S^1$. The sugra solution will contain three harmonic forms and hopefully a constant dilation background. See this review by Gautam Mandal discuss the D1-D5 system. You need to add another set of D5-branes to that solution.
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