The work is dealing with spatially-periodic solutions to the generalized (two-component) Hunter-Saxton (HS) equation, in the form of
This system is a short-wave (high-frequency) limit of the Camassa-Holm two-component system, which, in turn, is a model for non-small water waves, that admits wave breaking. On the other hand, the single-component HS equation, which has other physical realizations, in addition to the water waves, is integrable. The paper at first addresses the well-posedness of the initial-value problem for the two-component HS system in the spatially-periodic setting. Then, the wave-breaking criterion for particular initial profiles is established. The blow-up rate for strong wave-breaking solutions is considered too. Finally, sufficient conditions for the global existence of the solutions are given.