A Lax Pair Structure for the Half-Wave Maps Equation

We consider the half-wave maps equation $${\mathrm{\partial }}_t\vec{S}=\vec{S}\wedge \mathrm{\mid }\mathrm{\nabla }\mathrm{\mid }\vec{S},$$ where $\vec{S}=\vec{S}(t,x)$ takes values on the two-dimensional unit sphere ${\mathbb{S}}^2$ and $x\in \mathbb{R}$ (real line case) or $x\in \mathbb{T}$ (periodic case). This an energy-critical Hamiltonian evolution equation recently introduced in \cite{LS,Zh}, which formally arises as an effective evolution equation in the classical and continuum limit of Haldane-Shastry quantum spin chains. We prove that the half-wave maps equation admits a Lax pair and we discuss some analytic consequences of this finding. As a variant of our arguments, we also obtain a Lax pair for the half-wave maps equation with target ${\mathbb{H}}^2$ (hyperbolic plane).