Title: Kulish-Sklyanin type models: integrability and reductions,
Abstract: We start with a Riemann-Hilbert problem (RHP) related to a BD.I-type
symmetric spaces $SO(2r+1)/S(O(2r-2s +1)\otimes O(2s))$, $s\geq 1$. We consider
two Riemann-Hilbert problems: the first formulated on the real axis
$\mathbb{R}$ in the complex $\lambda$-plane; the second one is formulated on
$\mathbb{R} \oplus i\mathbb{R}$. The first RHP for $s=1$ allows one to solve
the Kulish-Sklyanin (KS) model; the second RHP is relevant for a new type of KS
model. An important example for nontrivial deep reductions of KS model is
given. Its effect on the scattering matrix is formulated. In particular we
obtain new 2-component NLS equations. Finally, using the Wronskian relations we
demonstrate that the inverse scattering method for KS models may be understood
as a generalized Fourier transforms. Thus we have a tool to derive all their
fundamental properties, including the hierarchy of equations and the hierarchy
of their Hamiltonian structures.