  We study an equation lying `mid-way' between the periodic Hunter-Saxton and
Camassa-Holm equations, and which describes evolution of rotators in liquid
crystals with external magnetic field and self-interaction. We prove that it is
an Euler equation on the diffeomorphism group of the circle corresponding to a
natural right-invariant Sobolev metric. We show that the equation is
bihamiltonian and admits both cusped, as well as smooth, traveling-wave
solutions which are natural candidates for solitons. We also prove that it is
locally well-posed and establish results on the lifespan of its solutions.
Throughout the paper we argue that despite similarities to the KdV, CH and HS
equations, the new equation manifests several distinctive features that set it
apart from the other three.
