  Equilibrium conditions and spin dynamics of spinning test particles are
discussed in the stationary and axially symmetric Kerr-de Sitter black-hole or
naked-singularity spacetimes. The general equilibrium conditions are
established, but due to their great complexity, the detailed discussion of the
equilibrium conditions and spin dynamics is presented only in the simple and
most relevant cases of equilibrium positions in the equatorial plane and on the
symmetry axis of the spacetimes. It is shown that due to the combined effect of
the rotation of the source and the cosmic repulsion the equilibrium is spin
dependent in contrast to the spherically symmetric spacetimes. In the
equatorial plane, it is possible at the so-called static radius, where the
gravitational attraction is balanced by the cosmic repulsion, for the spinless
particles as well as for spinning particles with arbitrarily large
azimuthal-oriented spin or at any radius outside the ergosphere with a
specifically given spin orthogonal to the equatorial plane. On the symmetry
axis, the equilibrium is possible at any radius in the stationary region and is
given by an appropriately tuned spin directed along the axis. At the static
radii on the axis the spin of particles in equilibrium must vanish.
