  The generalized, two-component Hunter-Saxton system comprises several
well-known models of fluid dynamics and serves as a tool for the study of
one-dimensional fluid convection and stretching. In this article a general
representation formula for periodic solutions to the system, which is valid for
arbitrary values of parameters $(\lambda,\kappa)\in\mathbb{R}\times\mathbb{R}$,
is derived. This allows us to examine in great detail qualitative properties of
blow-up as well as the asymptotic behaviour of solutions, including convergence
to steady states in finite or infinite time.
