We consider the half-wave maps equation
$$\begin{aligned} \partial _t \vec {S} = \vec {S} \wedge |\nabla | \vec {S}, \end{aligned}$$
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 Lenzmann and Schikorra (
2017,
arXiv:1702.05995v2), Zhou and Stone (Phys Lett A 379:2817–2825,
2015) 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).