Type IIB Flows with N=1 Supersymmetry

We write general and explicit equations which solve the supersymmetry transformations with two arbitrary complex-proportional Weyl spinors on $\mathcal{N}=1$ supersymmetric type IIB strings backgrounds with all R-R $F_1$, $F_3$, $F_5$ and NS-NS $H_3$ fluxes turned on using SU(3) structures. The equations are generalizations of the ones found for specific relations between the two spinors by Grana, Minasian, Petrini and Tomasiello in [1] and by Butti, Grana, Minasian, Petrini and Zaffaroni in [2]. The general equations allow to study systematically generic type IIB backgrounds with $\mathcal{N}=1$ supersymmetry. We then explore some specific classes of flows with constant axion, flows with constant dilaton, flows on conformally Calabi-Yau backgrounds, flows with imaginary self-dual 3-form flux, flows with constant ratio of the two spinors, the corresponding equations are written down and some of their features and relations are discussed.