  Brief review of the methods for solving the multicomponent nonlinear
Schrodinger (MNLS) equations and analysis of their Hamiltonian structures is
given. Main attention is paid to the MNLS related to the C.II- and D.III-types
symmetric spaces with nonvanishing (constant) boundary conditions. The spectral
properties of their Lax operators are described. The derivation of the trace
identities is outlined. The involutivity of their integrals of motion is proved
using the method of the classical R-matrix.
