  We present four examples of integrable partial differential equations (PDEs)
of mathematical physics that---when linearized around a stationary
soliton---exhibit scattering without reflection at {\it all} energies. Starting
from the most well-known and the most empirically relevant phenomenon of the
transparency of one-dimensional bright bosonic solitons to Bogoliubov
excitations, we proceed to the sine-Gordon, Korteweg-de Vries, and Liouville's
equation whose stationary solitons also support our assertion. The proposed
connection between integrability and reflectionless scattering seems to span at
least two distinct paradigms of integrability: S-integrability in the first
three cases, and C-integrability in the last one. We argue that the
transparency of linearized integrable PDEs is necessary to ensure that they can
support the transparency of stationary solitons in the original integrable
PDEs. As contrasting cases, the analysis is further extended to cover two
non-integrable systems: a sawtooth-Gordon and a $\phi^4$ model.
