First-Order Multipliers, Noether Symmetries and Variational Reduction of the Hunter–Saxton Equation

We revisit the Hunter–Saxton equation through its classical first-order Lagrangian. The determining system for all first-order multipliers is reduced to one linear equation in two variables. Its solutions include both point-symmetry characteristics and genuinely generalised variational characteristics; in the analytic category, the remaining freedom is locally parameterised by two arbitrary analytic functions. Every member of the known infinite-dimensional point-symmetry ideal is shown to preserve the action up to a total divergence and produces an arbitrary-function family of conserved currents, whereas one finite point symmetry is excluded from the Noether point-symmetry algebra of the Lagrangian. We also correct an omission in the previously reported associations between finite currents and Lie point symmetries. For the scaling symmetry, the radial component of an associated current vanishes identically after transformation. Alignment and multiplier criteria explain this degeneracy and show when it is a property of the conservation-law class. Thus association guarantees invariance of the transformed component, but not that the component retains differential content. Reducing the Lagrangian instead recovers the known similarity equations and their first integrals for two representative symmetries. In the scaling reduction, the reduced Noether symmetry is induced by a commuting member of the infinite-dimensional ideal.