  This is a review of two of the fundamental tools for analysis of soliton
equations: i) the algebraic ones based on Kac-Moody algebras, their central
extensions and their dual algebras which underlie the Hamiltonian structures of
the NLEE; ii) the construction of the fundamental analytic solutions of the Lax
operator and the Riemann-Hilbert problem (RHP) which they satisfy. The fact
that the inverse scattering problem for the Lax operator can be viewed as a RHP
gave rise to the dressing Zakharov-Shabat, one of the most effective ones for
constructing soliton solutions. These two methods when combined may allow one
to prove rigorously the results obtained by the abstract algebraic methods.
They also allow to derive spectral decompositions for non-self-adjoint Lax
operators.
