  This is the second part of a series of papers dealing with an extensive class
of analytic difference operators admitting reflectionless eigenfunctions. In
the first part, the pertinent difference operators and their reflectionless
eigenfunctions are constructed from given ``spectral data'', in analogy with
the IST for reflectionless Schr\"odinger and Jacobi operators. In the present
paper, we introduce a suitable time dependence in the data, arriving at
explicit solutions to a nonlocal evolution equation of Toda type, which may be
viewed as an analog of the KdV and Toda lattice equations for the latter
operators. As a corollary, we reobtain various known results concerning
reflectionless Schr\"odinger and Jacobi operators. Exploiting a
reparametrization in terms of relativistic Calogero--Moser systems, we also
present a detailed study of $N$-soliton solutions to our nonlocal evolution
equation.
