  We use geometric methods of equivariant dynamical systems to address a
long-standing open problem in the theory of nematic liquid crystals, namely a
proof of the existence and asymptotic stability of kayaking periodic orbits for
which the principal axis of orientation of the molecular field (the director)
rotates around the vorticity axis in response to steady shear flow. With a
small parameter attached to the symmetric part of the velocity gradient, the
problem can be viewed as a symmetry-breaking bifurcation from an orbit of the
rotation group~$\SO(3)$ that contains both logrolling (equilibrium) and
tumbling (periodic rotation of the director within the plane of shear) regimes
as well as a continuum of kayaking orbits. The results turn out to require
expansion to second order in the perturbation parameter.
