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In this work we have developed the essential tools for the algebraic Betheansatz solution of integrable vertex models invariant by a unique U(1) chargesymmetry. The formulation is valid for arbitrary statistical weights andrespective number $N$ of edge states. We show that the fundamental commutationrules between the monodromy matrix elements are derived by solving linearsystems of equations. This makes possible the construction of the transfermatrix eigenstates by means of a new recurrence relation depending on $N-1$distinct types of creation fields. The necessary identities to solve theeigenvalue problem are obtained exploring the unitarity property and theYang-Baxter equation satisfied by the $R$-matrix. The on-shell and off-shellproperties of the algebraic Bethe ansatz are explicitly presented in terms ofthe arbitrary $R$-matrix elements. This includes the transfer matrixeigenvalues, the Bethe ansatz equations and the structure of the vectors notparallel to the eigenstates.

Algebraic Bethe Ansatz for U(1) Invariant Integrable Models: The Method and General Results