This project focuses on prototypical integrable systems, their associated linear operators and scattering data as well as instability properties in inverse problems. To this end, firstly, we concentrate on the possibility of transferring bounds from integrable model systems to limiting dispersive equations, e.g., by deducing the Benjamin-Ono equation as a limit of the Calogero-Moser equation. Secondly, we will consider the role of the Bäcklund transform in critical L2-based spaces for adding and removing solitons. Thirdly, we will explore (in-)stability properties in inverse problems and establish a general framework for systems displaying threshold behaviour.