  We construct elliptic multi-soliton solutions of the spin non-chiral
intermediate long-wave (sncILW) equation with periodic boundary conditions.
These solutions are obtained by a spin-pole ansatz including a dynamical
background term; we show that this ansatz solves the periodic sncILW equation
provided the spins and poles satisfy the elliptic $A$-type spin Calogero-Moser
(sCM) system with certain constraints on the initial conditions. The key to
this result is a B\"{a}cklund transformation for the elliptic sCM system which
includes a non-trivial dynamical background term. We also present solutions of
the sncILW equation on the real line and of the spin Benjamin-Ono equation
which generalize previously obtained solutions by allowing for a non-trivial
background term.
