  We discuss a new solution of the Kompaneets-equation for physical situations
in which low frequency photons, forming relatively narrow spectral details, are
Compton scattered in an isotropic, infinite medium with an intense ambient
blackbody field that is very close to full thermodynamic equilibrium with the
free electrons. In this situation the background-induced stimulated Compton
scattering slows down the motion of photons toward higher frequencies by a
factor of 3 in comparison with the solution that only takes into account
Doppler-broadening and boosting. This new solution is important for detailed
computations of cosmic microwave background spectral distortions arising due to
uncompensated atomic transitions of hydrogen and helium in the early Universe.
In addition we derive another analytic solution that only includes the
background-induced stimulated Compton scattering and is valid for power-law
ambient radiation fields. This solution might have interesting applications for
radio lines arising inside of bright extra-galactic radio source, where
according to our estimates line shifts because of background-induced stimulated
scattering could be amplified and even exceed the line broadening due to the
Doppler-effect.
