  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.
