  In this work we have developed the essential tools for the algebraic Bethe
ansatz solution of integrable vertex models invariant by a unique U(1) charge
symmetry. The formulation is valid for arbitrary statistical weights and
respective number $N$ of edge states. We show that the fundamental commutation
rules between the monodromy matrix elements are derived by solving linear
systems of equations. This makes possible the construction of the transfer
matrix eigenstates by means of a new recurrence relation depending on $N-1$
distinct types of creation fields. The necessary identities to solve the
eigenvalue problem are obtained exploring the unitarity property and the
Yang-Baxter equation satisfied by the $R$-matrix. The on-shell and off-shell
properties of the algebraic Bethe ansatz are explicitly presented in terms of
the arbitrary $R$-matrix elements. This includes the transfer matrix
eigenvalues, the Bethe ansatz equations and the structure of the vectors not
parallel to the eigenstates.
