  We calculate explicitly the Bethe vectors states by the algebraic Bethe
ansatz method with the $gl(2)$-invariant $R$-matrix for the two-site
Bose-Hubbard model. Using a binomial expansion of the n-th power of a sum of
two operators we get and solve a recursion equation. We calculate the scalar
product and the norm of the Bethe vectors states. The form factors of the
imbalance current operator are also computed.
