FLEX method was invented by N.E.Bickers and D.J.Scalapino in 1989 (PRL 62,961; Ann. Phys. 193,206). Later it was extended to multi-orbital system (T.Takimoto,PRB,69,104504).
But I don't find FLEX including the SOC term. More generally, can FLEX method be applied to the $SU(2)$ spin-rotation symmetry broken Hamiltonian? And does any one know something about it?