In mathematics, the Gelfand–Zeitlin system (also written Gelfand–Zetlin system, Gelfand–Cetlin system, Gelfand–Tsetlin system) is an integrable system on conjugacy classes of Hermitian matrices. It was introduced by Guillemin and Sternberg (1983), who named it after the Gelfand–Zeitlin basis, an early example of canonical basis, introduced by I. M. Gelfand and M. L. Cetlin in 1950s. Kostant and Wallach (2006) introduced a complex version of this integrable system.

• Guillemin, Victor; Sternberg, Shlomo (1983), "The Gel'fand-Cetlin system and quantization of the complex flag manifolds", Journal of Functional Analysis, 52 (1): 106–128, doi:10.1016/0022-1236(83)90092-7, ISSN 0022-1236, MR 0705993
• Kostant, Bertram; Wallach, Nolan (2006), "Gelfand-Zeitlin theory from the perspective of classical mechanics. I", Studies in Lie theory, Progr. Math., vol. 243, Boston, MA: Birkhäuser Boston, pp. 319–364, arXiv:math/0408342, doi:10.1007/0-8176-4478-4_12, ISBN 978-0-8176-4342-3, MR 2214253
• Kogan, Mikhail; Miller, Ezra (2005). "Toric degeneration of Schubert varieties and Gelfand—Tsetlin polytopes". Advances in Mathematics. 193 (1): 1–17. doi:10.1016/j.aim.2004.03.017.