number \\
& 6R^{[ab|e|}Q^{cd]}_e+{F}_a P^{a,abcd}+{F}_b P^{b,abcd}+{F}_c P^{c,abcd}+{F}_d P^{d,abcd}+\tfrac{1}{2}P^{a,bcdaef}{F}_{aef}\nonumber \\
& \qquad \qquad +\tfrac{1}{2}P^{b,bcdaef}{F}_{bef}+\tfrac{1}{2}P^{c,bcdaef}{F}_{cef}+\tfrac{1}{2}P^{d,bcdaef}{F}_{def} =0 \nonumber \end{aligned}$$ in the IIB case, and $$\begin{aligned}
&  6f^{e}_{[ab}H_{cd]e}+4P^e_{[a}F_{bcd]e}=0 \nonumber \\
& 3Q^{ae}_{[b}H_{cd]e}+3f^{e}_{[bc}f^{a}_{d]e}-3P^{a}_{[b}F_{cd]}-P^{a,a}F_{bcda}+\tfrac{1}{2}P^{a,aef}F_{bcdaef}+\tfrac{3}{2} P^{aef}_{[b}F_{cd]ef}=0  \nonumber \\
&-Q^{ab}_e f^e_{cd}-4Q^{[a|e|}_{[c}f^{b]}_{d]e}-R^{abe}H_{cde}
+2P^{abe}_{[c}{F}_{d]e}-P^{a,abe}{F}_{cdae}-P^{b,abe}{F}_{cdbe} \nonumber \\
& \qquad \qquad  -\tfrac{1}{3} P^{abefg}_{[c} F_{d]efg} + \tfrac{1}{6} P^{a,abefg} F_{cdaefg} +\tfrac{1}{6} P^{b,abefg} F_{cdbefg} =0 \label{NSNSBianchiIIAwithP}  \\
& 3R^{[ab|e|}f^{c]}_{de}+3Q^{[ab}_eQ^{c]e}_d+ {F} P^{abc}_d-P^{a,abc}{F}_{ad}-P^{b,abc}{F}_{bd}-P^{c,abc}{F}_{cd} \nonumber \\
& \qquad \qquad -\tfrac{1}{2} P^{abcef}_d {F}_{ef}- \tfrac{1}{2} P^{a,bcaef}{F}_{daef}- \tfrac{1}{2} P^{b,bcaef}{F}_{dbef}- \tfrac{1}{2} P^{c,bcaef}{F}_{dcef}=0\nonumber \\
& 6R^{[ab|e|}Q^{cd]}_e-{F}_{de}P^{d,abcde}-{F}_{ce}P^{c,abcde}-{F}_{be}P^{b,abcde}-{F}_{ae}P^{a,abcde}=0 \nonumber \end{aligned}$$ in the IIA case.

The NS-NS quadratic constraints can be relaxed by the inclusion of sources [@Villadoro:2007tb; @andriot]. This obviously also applies to the constraints modified by the inclusion of $P$ fluxes in eqs. and . In particular, relaxing the first constraints (that we schematically write as $( {\rm flux} \cdot {\rm flux}  )_4 =0$) in both equations induces a charge for the NS5-brane coming from the generalised Chern-Simons term $$\int D_6 \wedge ( {\rm flux} \cdot {\rm flux} )_4 \quad .$$ As we have discussed in section 3, if we now consider a particular component for $D_6$, and we perform a T-duality $T_a$ in a direction whose index $a$ is not contained in $D_6$, this is mapped to $D_{6 \, a,a}$ which is a component of the mixed-sy