  The isospectral flows of an $n^{th}$ order linear scalar differential
operator $L$ under the hypothesis that it possess a Baker-Akhiezer function
were originally investigated by Segal and Wilson from the point of view of
infinite dimensional Grassmanians, and the reduction of the KP hierarchy to the
Gel'fand-Dikii hierarchy. The associated first order systems and their formal
asymptotic solutions have a rich Lie algebraic structure which was investigated
by Drinfeld and Sokolov. We investigate the matrix Riemann-Hilbert
factorizations for these systems, and show that different factorizations lead
respectively to the potential, modified, and ordinary Gel'fand-Dikii flows. Lie
algebra decompositions (the Adler-Kostant-Symes method) are obtained for the
modified and potential flows. For $n>3$ the appropriate factorization for the
Gel'fand-Dikii flows is not a group factorization, as would be expected; yet a
modification of the dressing method still works.
  A direct proof, based on a Fredholm determinant associated with the
factorization problem, is given that the potentials are meromorphic in $x$ and
in the time variables. Potentials with Baker-Akhiezer functions include the
multisoliton and rational solutions, as well as potentials in the scattering
class with compactly supported scattering data. The latter are dense in the
scattering class.
