  This work is concerned with Mishchenko-Fomenko subalgebras and their
restrictions to the adjoint orbits in a finite-dimensional complex semisimple
Lie algebra. In this setting, it is known that each Mishchenko-Fomenko
subalgebra restricts to a completely integrable system on every orbit in
general position. We improve upon this result, showing that each
Mishchenko-Fomenko subalgebra yields a completely integrable system on all
regular orbits (i.e. orbits of maximal dimension). Our approach incorporates
the theory of regular $\mathfrak{sl}_2$-triples and associated Slodowy slices,
as developed by Kostant.
