  The stellar spin orientation relative to the orbital planes of multiplanet
systems are becoming accessible to observations. Here, we analyze and classify
different types of spin-orbit evolution in compact multiplanet systems
perturbed by an inclined outer companion. Our study is based on classical
secular theory, using a vectorial approach developed in a separate paper. When
planet-planet perturbations are truncated at the second order in eccentricity
and mutual inclination, and the planet-companion perturbations are developed at
the quadrupole order, the problem becomes integrable. The motion is composed of
a uniform precession of the whole system around the total angular momentum, and
in the rotating frame, the evolution is periodic. Here, we focus on the
relative motion associated to the oscillations of the inclination between the
planet system and the outer orbit, and of the obliquities of the star with
respect to the two orbital planes. The solution is obtained using a powerful
geometric method. With this technique, we identify four different regimes
characterized by the nutation amplitude of the stellar spin-axis relative to
the orbital plane of the planets. In particular, the obliquity of the star
reaches its maximum when the system is in the Cassini regime where planets have
more angular momentum than the star, and where the precession rate of the star
is similar to that of the planets induced by the companion. In that case,
spin-orbit oscillations exceed twice the inclination between the planets and
the companion. Even if mutual inclination is only ~ 20 deg, this resonant case
can cause the spin-orbit angle to oscillate between perfectly aligned and
retrograde values.
