Three decades ago, Inozemtsev discovered an isotropic long-range spin chain with elliptic pair potential that interpolates between the Heisenberg and Haldane–Shastry spin chains while admitting an exact solution throughout, based on a connection with the elliptic quantum Calogero–Sutherland model. Though Inozemtsev’s spin chain is widely believed to be quantum integrable, the underlying algebraic reason for its exact solvability is not yet well understood. As a step in this direction we refine Inozemtsev’s ‘extended coordinate Bethe ansatz’ and clarify various aspects of the model’s exact spectrum and its limits. We identify quasimomenta in terms of which the M-particle energy is close to being (functionally) additive, as one would expect from the limiting models. This moreover makes it possible to rewrite the energy and Bethe-ansatz equations on the elliptic curve, turning the spectral problem into a rational problem as might be expected for an isotropic spin chain. We treat the M=2 particle sector and its limits in detail. We identify an S-matrix that is independent of positions despite the more complicated form of the extended coordinate Bethe ansatz. We show that the Bethe-ansatz equations reduce to those of Heisenberg in one limit and give rise to the ‘motifs’ of Haldane–Shastry in the other limit. We show that, as the interpolation parameter changes, the ‘scattering states’ from Heisenberg become Yangian highest-weight states for Haldane–Shastry, while bound states become (sl₂-highest weight versions of) affine descendants of the magnons from M=1. We are able to treat this at the level of the wave function and quasimomenta. For bound states we find an equation that, for given Bethe integers, relates the ‘critical’ values of the spin-chain length and the interpolation parameter for which the two complex quasimomenta collide; it reduces to the known equation for the ‘critical length’ in the limit of the Heisenberg spin chain. We also elaborate on Inozemtsev’s proof of the completeness for M=2 by passing to the elliptic curve. Our review of the two-particle sectors of the Heisenberg and Haldane–Shastry spin chains may be of independent interest.

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 Bäcklund transformation of this spin CM system, generalize our results to the periodic HWM equation, and provide plots that visualize our soliton solutions.