  We study the spin-charge coupled transport in a two-dimensional electron
system using the method of quasiclassical ($\xi$-integrated) Green's functions.
In particular we derive the Eilenberger equation in the presence of a generic
spin-orbit field. The method allows us to study spin and charge transport from
ballistic to diffusive regimes and continuity equations for spin and charge are
automatically incorporated. In the clean limit we establish the connection
between the spin-Hall conductivity and the Berry phase in momentum space. For
finite systems we solve the Eilenberger equation numerically for the special
case of the Rashba spin-orbit coupling and a two-terminal geometry. In
particular, we calculate explicitly the spin-Hall induced spin polarization in
the corners, predicted by Mishchenko et al. [13]. Furthermore we find universal
spin currents in the short-time dynamics after switching on the voltage across
the sample, and calculate the corresponding spin-Hall polarization at the
edges. Where available, we find perfect agreement with analytical results.
