  We implement our new Separation of Variables (SoV) approach for open quantum
integrable models associated to higher rank representations of the reflection
algebras. We construct the (SoV) basis for the fundamental representations of
the $Y(gl_n)$ reflection algebra associated to general integrable boundary
conditions. Moreover, we give the conditions on the boundary $K$-matrices
allowing for the transfer matrix to be diagonalizable with simple spectrum.
These SoV basis are then used to completely characterize the transfer matrix
spectrum for the rank one and two reflection algebras. The rank one case is
developed for both the rational and trigonometric fundamental representations
of the 6-vertex reflection algebra. On the one hand, we extend the complete
spectrum characterization to representations previously lying outside the SoV
approach, e.g. those for which the standard Algebraic Bethe Ansatz applies. On
the other hand, we show that our new SoV construction can be reduced to the
generalized Sklyanin's one whenever it is applicable. The rank two case is
developed explicitly for the fundamental representations of the $Y(gl_3)$
reflection algebra associated to general integrable boundary conditions. For
both rank one and two our SoV approach leads to a complete characterization of
the transfer matrix spectrum in terms of a set of polynomial solutions to the
corresponding quantum spectral curve equation. Those are finite difference
functional equations of order equal to the rank plus one, i.e., here two and
three respectively for the $Y(gl_2)$ and $Y(gl_3)$ reflection algebras.
