Fundamental analytic solutions for the Kulish-Sklyanin model with constant boundary conditions

In the present paper we analyze construction of fundamental analytic solutions (FAS) for generalized Kulish-Sklyanin models (KSM) vanishing (VBC) and constant boundary conditions (CBC). Using FAS one can reduce direct inverse scattering problems Lax operator to a Riemann-Hilbert problem (RHP). For VBC find two $\chi^+(x,t,\lambda) $ $\chi^-(x,t,\lambda) in upper/lower $\mathbb{C}_\pm$ complex $\lambda$-plane. The RHP consists in: given sewing function $G(x,t,\lambda)$ constructing both $\chi^\pm(x,t,\lambda) their regions analyticity. CBC becomes more complicated, because now must be formulated on Riemannian surface genus 1.