Integrability structures of the generalized Hunter-Saxton equation

I will talk about integrability structures of the generalized Hunter-Saxton equation. In particular, I will present a Lax representation with non-removable parameter, local recursion operators of the first order for symmetries and cosymmetries, and a Lie algebra of higher symmetries generated by the recursion operators from contact symmetries. Also I will show an example of constructing solutions invariant with respect to a higher symmetry.