e resulting superpotential of each theory is mapped to the one of the other theory provided that the moduli $U$ and $T$ are interchanged and the $P$ fluxes in the two theories are identified according to our rule, which amounts to the statement that the duality between the IIB and IIA model is restored. The IIB expression for the superpotential coincides with the expression of [@Aldazabal:2010ef] as far as the $P$ fluxes are concerned.

Turning on fluxes in general modifies the Bianchi identities for the field strengths of the various potentials in the theory in such a way to produce an effective charge for these potentials which is given by the integral in the internal directions of a quadratic term in the fluxes. In particular, the absence of NS branes leads to a set of quadratic constraints for all the NS-NS fluxes [@Shelton:2005cf; @Aldazabal:2006up; @wrase; @penas; @shukla1; @andriot; @shukla2]. By using our T-duality rules, we will manage to show how these constraints are modified by the inclusion of terms containing both RR and $P$ fluxes. The RR Bianchi identities, instead, can give a non-vanishing contribution to the effective number of D-branes, which has to be taken into account in the tadpole cancellation condition. In IIB, turning on the $P_a^{bc}$ flux leads to a new tadpole that is cancelled by the addition of a proper number of 7-branes which are the S-duals of the D7-branes [@Aldazabal:2006up]. By turning on more general $P$ fluxes, one then expects that additional branes, which are related by suitable T-duality transformations to the S-dual of the D7-brane, can be included.

In a series of papers [@Bergshoeff:2010xc; @stringsolitons; @axel; @dominantweights; @bergshoeffriccionimarrani], all the 1/2-BPS branes that are present in string theory in any dimension have been classified according to their properties with respect to the duality symmetry group of the theory. By carefully analysing this classification, we will manage to determine a universal T-duality transformation rule for all the bra