  We give explicit solutions to the two-component Hunter-Saxton system on the
unit circle. Moreover, we show how global weak solutions can be naturally
constructed using the geometric interpretation of this system as a
re-expression of the geodesic flow on the semi-direct product of a suitable
subgroup of the diffeomorphism group of the circle with the space of smooth
functions on the circle. These spatially and temporally periodic solutions turn
out to be conservative.
