  We suggest a generalization of nonlinear $\sigma$-model for diffusive
superconducting systems to account for magnetoelectric effects due to
spin-orbit scattering. In the leading orders of spin-orbit strength and
gradient expansion it includes two additional terms responsible for the
spin-Hall effect and the spin-current swapping. First, assuming a
delta-correlated disorder we derive the new terms from the Keldysh path
integral representation of the generating functional. Then we argue
phenomenologically that they exhaust all invariants allowed in the effective
action to the leading order in the spin-orbit coupling (SOC). Finally, the
results are confirmed by a direct derivation of the saddle-point (Usadel)
equation from the quantum kinetic equations in the presence of randomly
distributed impurities with SOC. At this point we correct a recent derivation
of the Usadel equation that includes magneto-electric effects and does not
resort to the Born approximation.
