  We consider the half-wave maps (HWM) equation which provides a continuum
description of the classical Haldane-Shastry spin chain on the real line. We
present exact multi-soliton solutions of this equation. Our solutions describe
solitary spin excitations that can move with different velocities and interact
in a non-trivial way. We make an ansatz for the solution allowing for an
arbitrary number of solitons, each described by a pole in the complex plane and
a complex spin variable, and we show that the HWM equation is satisfied if
these poles and spins evolve according to the dynamics of an exactly solvable
spin Calogero-Moser (CM) system with certain constraints on initial conditions.
We also find first order equations providing a B\"acklund transformation of
this spin CM system, generalize our results to the periodic HWM equation, and
provide plots that visualize our soliton solutions.
