Equations of motion for Earth and Moon
I may have found a partial answer to my question in the Navigational Astronomy chapter of 'American Practical Navigator' by N. Bowditch . On page 228 


  [...] nutation, an irregularity in the motion of the Earth due to the
  disturbing effect of other celestial bodies, principally the Moon


and on page 230 


  The precession of the Earth’s axis is the result of gravitational
  forces exerted principally by the Sun and Moon on the Earth’s
  equatorial bulge. The spinning Earth responds to these forces in the
  manner of a gyroscope. Regression of the nodes introduces certain
  irregularities known as nutation in the precessional motion.


So the dominant effects as for predicting the orientation of the Earth's axis seem to be the Earth's ellipticity combined with the attraction of the Sun and Moon.