Summary: A new system of equations generalizing the Hunter-Saxton system [the author, Discrete Contin. Dyn. Syst., Ser. B 12, No. 3, 647--656 (2009; Zbl 1176.35028
)] is derived and investigated. In the periodic setting, we show that there exist local solutions for which we provide sufficient conditions in order that they stay bounded or blow up in finite time; on the real line, we construct a class of global weak solutions.