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Thanks to results of A. Beauville (Acta Math. 164 (1990) 211–235), we give an explicit description of translation-invariant vector fields on affine Jacobians of spectral curves, which gives a mathematical support for the work of F.A. Smirnov and V. Zeitlin (preprint math-ph/0203037). In the hyperelliptic case, this description is due to D. Mumford (Tata Lectures on Theta, Vol. II, Birkha¨user, Boston, 1983). To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003). [Copyright &y& Elsevier]