… We provide a rigorous treatment of the inverse scattering transform for the entire Toda
hierarchy for solutions which are asymptotically close to (in general) different finite-gap
solutions as n → ± ∞ ⁠. … Our aim here is to fill this gap and to provide a treatment of the
inverse scattering transform for the entire Toda hierarchy in the case of steplike
quasiperiodic finite-gap backgrounds. Note that since we treat the entire Toda hierarchy,
our results also cover the Kac–van Moerbeke hierarchy as a special case. …