  We consider integrability structures of the generalized Hunter--Saxton
equation. In particular, we obtain the Lax representation with nonremovable
spectral parameter, find local recursion operators for symmetries and
cosymmetries, generate an infinite-dimensional Lie algebra of higher
symmetries, and prove existence of infinite number of cosymmetries of higher
order. Further, we give an example of employing the higher order symmetry to
constructing exact globally defined solutions for the generalized
Hunter--Saxton equation.
