  We extend our previous work devoted to the computation of the
next-to-next-to-leading order spin-orbit correction (corresponding to 3.5PN
order) in the equations of motion of spinning compact binaries, by: (i)
Deriving the corresponding spin-orbit terms in the evolution equations for the
spins, the conserved integrals of the motion and the metric regularized at the
location of the particles (obtaining also the metric all-over the near zone but
with some lower precision); (ii) Performing the orbital reduction of the
precession equations, near-zone metric and conserved integrals to the center-
of-mass frame and then further assuming quasi-circular orbits (neglecting
gravitational radiation reaction). The results are systematically expressed in
terms of the spin variables with conserved Euclidean norm instead of the
original antisymmetric spin tensors of the pole-dipole formalism. This work
paves the way to the future computation of the next-to-next-to-leading order
spin-orbit terms in the gravitational-wave phasing of spinning compact
binaries.
