  We study dc-transport and magnetization dynamics in a junction of arbitrary
transparency consisting of two spin-singlet superconducting leads connected via
a single classical spin precessing at the frequency $\Omega$. The presence of
the spin in the junction provides different transmission amplitudes for spin-up
and spin-down quasiparticles as well as a time-dependent spin-flip transmission
term. For a phase biased junction, we show that a steady-state superconducting
charge current flows through the junction and that an out-of-equilibrium
circularly polarized spin current, of frequency $\Omega$, is emitted in the
leads. Detailed understanding of the charge and spin currents is obtained in
the entire parameter range. In the adiabatic regime, $\hbar \Omega \ll 2\Delta$
where $\Delta$ is the superconducting gap, and for high transparencies of the
junction, a strong suppression of the current takes place around $\vp \approx
0$ due to an abrupt change in the occupation of the Andreev bound-states. At
higher values of the phase and/or precession frequency, extended
(quasi-particle like) states compete with the bound-states in order to carry
the current. Well below the superconducting transition, these results are shown
to be weakly affected by the back-action of the spin current on the dynamics of
the precessing spin. Indeed, we show that the Gilbert damping due to the
quasi-particle spin current is strongly suppressed at low-temperatures, which
goes along with a shift of the precession frequency due to the condensate. The
results obtained may be of interest for on-going experiments in the field of
molecular spintronics.
