  We compute next-to-next-to-leading order spin contributions to the
post-Newtonian equations of motion for binaries of compact objects, such as
black holes or neutron stars. For maximally spinning black holes, those
contributions are of third-and-a-half post-Newtonian (3.5PN) order, improving
our knowledge of the equations of motion, already known for non-spinning
objects up to this order. Building on previous work, we represent the rotation
of the two bodies using a pole-dipole matter stress-energy tensor, and iterate
Einstein's field equations for a set of potentials parametrizing the metric in
harmonic coordinates. Checks of the result include the existence of a conserved
energy, the approximate global Lorentz invariance of the equations of motion in
harmonic coordinates, and the recovery of the motion of a spinning object on a
Kerr background in the test-mass limit. We verified the existence of a contact
transformation, together with a redefinition of the spin variables that makes
our result equivalent to a previously published reduced Hamiltonian, obtained
from the Arnowitt-Deser-Misner (ADM) formalism.
