  A Lagrangian formalism is used to study the motion of a spinning massive
particle in Friedmann--Robertson--Walker and G\"odel spacetimes, as well as in
a general Schwarzschild-like spacetime and in static spherically symmetric
conformally flat spacetimes. Exact solutions for the motion of the particle and
general exact expressions for the momenta and velocities are displayed for
different cases. In particular, the solution for the motion in spherically
symmetric metrics is presented in the equatorial plane. The exact solutions are
found using constants of motion of the particle, namely its mass, its spin, its
angular momentum, and a fourth constant, which is its energy when the metric is
time independent, and a different constant otherwise. These constants are
associated to Killing vectors. In the case of the motion on the
Friedmann--Robertson--Walker metric, a new constant of motion is found. This is
the fourth constant which generalizes previously known results obtained for
spinless particles. In the case of general Schwarzschild-like spacetimes, our
results allow for the exploration of the case of the
Reissner-Nordstrom-(Anti)de Sitter metric. Finally, for the case of the
conformally flat spacetimes, the solution is explicitly evaluated for different
metric tensors associated to a universe filled with static perfect fluids and
electromagnetic radiation. For some combination of the values of the constants
of motion the particle trajectories may exhibit spacelike velocity vectors in
portions of the trajectories.
