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Biconservative surfaces are surfaces with divergence-free stress-bienergytensor. Simply connected, complete, non-$CMC$ biconservative surfaces in$3$-dimensional space forms were constructed working in extrinsic and intrinsicways. Then, one raises the question of the uniqueness of such surfaces. In thispaper we give a positive answer to this question.

On the uniqueness of complete biconservative surfaces in $3$-dimensional space forms