  In this paper we show how the Functional Separation of Variables (FSoV)
method can be applied to the problem of computing overlaps with integrable
boundary states in integrable systems. We demonstrate our general method on the
example of a particular boundary state, a singlet of the symmetry group, in an
su(3) rational spin chain in an alternating fundamental--anti-fundamental
representation. The FSoV formalism allows us to compute in determinant form not
only the overlaps of the boundary state with the eigenstates of the transfer
matrix, but in fact with any factorisable state. This includes off-shell Bethe
states, whose overlaps with the boundary state have been out of reach with
other methods. Furthermore, we also found determinant representations for
insertions of so-called Principal Operators (forming a complete algebra of all
observables) between the boundary and the factorisable state as well as certain
types of multiple insertions of Principal Operators. Concise formulas for the
matrix elements of the boundary state in the SoV basis and su(N)
generalisations are presented. Finally, we managed to construct a complete
basis of integrable boundary states by repeated action of conserved charges on
the singlet state. As a result, we are also able to compute the overlaps of all
of these states with integral of motion eigenstates.
