We analyze the algebraic aspects of solving the multicomponent nonlinear Schrödinger (MNLS) equations related to the symmetric spaces of BD.I-type. This analysis includes: i) the spectral properties of the MNLS equations under the nonvanishing (constant) boundary conditions; ii) the construction of new equations of MNLS type imposing additional reductions; iii) the involutivity of their integrals of motion proven by using the method of the classical R-matrix; iv) brief but explicit description of their hierarchies of Hamiltonian structures.