  We study integrable models solvable by the nested algebraic Bethe ansatz and
possessing $GL(3)$-invariant $R$-matrix. Assuming that the monodromy matrix of
the model can be expanded into series with respect to the inverse spectral
parameter, we define zero modes of the monodromy matrix entries as the first
nontrivial coefficients of this series. Using these zero modes we establish new
relations between form factors of the elements of the monodromy matrix. We
prove that all of them can be obtained from the form factor of a diagonal
matrix element in special limits of Bethe parameters. As a result we obtain
determinant representations for form factors of all the entries of the
monodromy matrix.
