  The quasiclassical theory in terms of equations for the Green's functions
(Eilenberger equations) is generalized in order to allow for quantitative
description of the magneto-electric effects and proximity-induced triplet
correlations in the presence of spin-orbit coupling in hybrid superconducting
systems. The formalism is valid under the condition that the spin-orbit
coupling is weak with respect to the Fermi energy, but exceeds the
superconducting energy scale considerably. On the basis of the derived
formalism it is shown that the triplet correlations in the spin-orbit coupled
normal metal can be induced by proximity to a singlet superconductor without
any exchange or external magnetic field. They contain an odd-frequency
even-momentum component, which is stable against disorder. The value of the
proximity-induced triplet correlations is of the order of
$\Delta_{so}/\varepsilon_F$, that is absent in the framework of the standard
quasiclassical approximation, but can be described by our theory. The spin
polarization, induced by the Josephson current flowing through the
superconductor/Rashba metal/superconductor junction, is also calculated.
