  The dispersionless Toda hierarchy turns out to lie in the heart of a recently
proposed Landau-Ginzburg formulation of two-dimensional string theory at
self-dual compactification radius. The dynamics of massless tachyons with
discrete momenta is shown to be encoded into the structure of a special
solution of this integrable hierarchy. This solution is obtained by solving a
Riemann-Hilbert problem. Equivalence to the tachyon dynamics is proven by
deriving recursion relations of tachyon correlation functions in the machinery
of the dispersionless Toda hierarchy. Fundamental ingredients of the
Landau-Ginzburg formulation, such as Landau-Ginzburg potentials and tachyon
Landau-Ginzburg fields, are translated into the language of the Lax formalism.
Furthermore, a wedge algebra is pointed out to exist behind the Riemann-Hilbert
problem, and speculations on its possible role as generators of ``extra''
states and fields are presented.
