We study the global existence of solutions to a two-component generalized
Hunter-Saxton system in the periodic setting. We first prove a persistence
result of the solutions. Then for some particular choices of parameters
$(\alpha, \kappa)$, we show the precise blow-up scenarios and the existence of
global solutions to the generalized Hunter-Saxton system under proper
assumptions on the initial data. This significantly improves recent results
obtained in [M. Wunsch, DCDS Ser. B 12 (2009), 647-656] and [M. Wunsch, SIAM J.
Math. Anal. 42 (2010), 1286-1304].