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

<jats:p>We revisit the Hunter–Saxton equation using its classical first-order Lagrangian. We reduce the determining system for all multipliers depending on (t,x,u,ut,ux) completely to a single linear equation for Φ(t,ux); the resulting family includes characteristics of both point and generalized variational symmetries. Within the known infinite-dimensional point-symmetry ideal Yf=f(t)∂x+f′(t)∂u, we show that every generator is a Noether divergence symmetry and derive its arbitrary-function family of conserved currents, while the finite generator X1 is excluded from the algebra of Noether point-divergence symmetries. We also correct the symmetry–conservation-law associations and remove trivial current terms. For the scaling generator X3=t∂t+x∂x, the canonical representative of its current has an identically vanishing transformed radial component. We derive alignment and multiplier criteria for this degeneracy; the latter proves that, among associated representatives, it is a property of the conservation-law equivalence class. Thus association guarantees invariance of the transformed component, not that it retains differential content. Finally, we reduce the Lagrangian itself in canonical coordinates for two representative variational symmetries. The reduced Euler–Lagrange equations reproduce the known similarity equations, while their first integrals follow directly from the reduced variational structure. For the scaling reduction, the reduced Noether symmetry is induced by the commuting ideal element Yt. The resulting analysis separates new structural results from previously known similarity solutions.</jats:p>