  We propose the new family of the exactly solvable discrete state BCS - type
Hamiltonians based on its relationship to the six-vertex model in the
quasiclassical limit both in the rational and the trigonometric cases. We
establish the relation of the BCS Hamiltonian and its eigenfunctions to the
form of the monodromy matrix in the F-basis. Using the Algebraic Bethe Ansatz
method for the standard BCS model with equal coupling the expression for the
general scalar product and the determinant expressions for the physically
interesting correlation functions for the finite number of sites which can be
used in the numerical and analytical computations are obtained. We also compare
the correlators with the results obtained by means of the variational method.
