  We study the integrable two-site states of the quantum integrable models
solvable by the nested algebraic Bethe ansatz and possessing
$\mathfrak{gl}(N)$-invariant R-matrix. We investigate the overlaps between the
integrable two-site states and the wave-functions. To find exact derivations
for the factorized overlap formulas for the nested integrable systems is a
longstanding unsolved problem. In this paper we give a derivation for a large
class of the integrable states of the $\mathfrak{gl}(N)$ symmetric spin chain.
The first part of the derivation is to calculate recurrence relations for the
off-shell overlap that uniquely fix it. Using these recursions we prove that
the normalized overlaps of the multi-particle states have factorized forms
which contain the products of the one-particle overlaps and the ratio of the
Gaudin-like determinants. We also show that the previously proposed overlap
formulas agree with our general formula.
