  We consider closed XXX spin chains with broken total spin $U(1)$ symmetry
within the framework of the modified algebraic Bethe ansatz. We study multiple
actions of the modified monodromy matrix entries on the modified Bethe vectors.
The obtained formulas of the multiple actions allow us to calculate the scalar
products of the modified Bethe vectors. We find an analog of Izergin-Korepin
formula for the scalar products. This formula involves modified Izergin
determinants and can be expressed as sums over partitions of the Bethe
parameters.
