Here are some references on Hamiltonian superintegrable systems. They are also known as non-commutative integrable systems and as systems with degenerate integrability and slides of my talk at BIMSA: talk1, talk 2.

Earlier physics references:

  • W. Pauli, On the hydrogen spectrum from the standpoint of the new quantum mechanics, Zeitschrift fur Physik, 36, 336-363 (1926).
  • W. Pauli. Z.Physik 36:336 (1935).
  • Fock, V., Zur Theorie Des Wasserstoffatoms, Z. Physik 98, 145 (1935)
  • J. Frish, V. Mandrosov, Y.A. Smorodinsky, M. Uhlir and P. Winternitz. On higher symmetries in quantum mechanics Physics Letters 16:354-356 (1965).

Early mathematical works:

  • N.N. Nekhoroshev. Action-angle variables and their generalizations. Trans. Moscow Math. Soc. 26:180- 197 (1972).
  • Mischenko A.S., Fomenko, A.T., Generalized Liouville method or integrating Hamiltonian systems, Funct. Analysis and Applications, 1978, v. 12, n. 2, 4656.

Books, conference proceedings, reviews:

  • Superintegrability in Classical and Quantum Systems, Edited by: P. Tempesta, P. Winternitz, J. Harnad, W. Miller, Jr., G. Pogosyan, M. Rodriguez, CRM Proceedings and Lecture Notes, Volume: 37, 2004.
  • N. Reshetikhin, Degenerately Integrable Systems, J Math Sci (2016) 213: 769.

Superintegrability of spin Calogero-Moser systems:

  • Gibbons J., Hermsen T., A generalization of the Calogero -Moser system., Physica, 11D(1984), 337.
  • L.C. Li, P. Xu, Spin Calogero-Moser systems associated with simple Lie algebras C.R.Acad. Sci. Paris, Serie I , 331: n1, 55-61(2000).
  • N. Reshetikhin, Degenerate integrability of the spin Calogero-Moser systems and the duality with the spin Ruijsenaars systems. Lett. Math. Phys. 63 (2003), no. 1, 5571.
  • Feher, L.; Pusztai, B. G., Twisted spin Sutherland models from quantum Hamiltonian reduction. J. Phys. A 41 (2008), no. 19, 194009.
  • Feher, L., Klimcik, C. , Poisson-Lie interpretation of trigonometric Ruijsenaars duality, Commun. Math. Phys. 301 (2011),55-104.
  • V. Ayadi, L. Feher, T.F. Gorbe; Superintegrability of rational Ruijsenaars-Schneider systems and their action-angle duals, J. Geom. Symmetry Phys. 27 (2012) 27-44.
  • Feher, L., Klimyk, C., Self-duality of the compactified Ruijsenaars-Schneider system from quasi- Hamiltonian reduction, Nucl.Phys. B860 (2012) 464-515.
  • Feher, L.; Pusztai, B. G., Generalized spin Sutherland systems revisited, Nucl. Phys. B893 (2015) 236- 256.
  • O. Chalykh, M. Fairon, On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system, arXiv:1811.08727.
  • S. Kharchev, A. Levin, M. Olshanetsky, A. Zotov, Quasi-compact Higgs bundles and CalogeroSutherland systems with two types of spins, J. Math. Phys., 59:10 (2018), 103509 , 36 pp., arXiv: 1712.08851.
  • N. Reshetikhin Spin Calogero-Moser models on symmetric spaces. arXiv:1903.03685.

Other examples of superintegrable systems:

  • M.I. Gekhtman, M.Z. Shapiro. Non-commutative and commutative integrability of generic Toda flow in simple Lie algberas. Comm. Pure Appl. Math. 52: 5384 (1999).
  • N. Reshetikhin, Integrability of characteristic Hamiltonian systems on simple Lie groups with standard Poisson Lie structure. Comm. Math. Phys. 242 (2003), no. 1-2, 129.
  • N.Reshetikhin, G. Schrader, Superintegrability of Generalized Toda Models on Symmetric Spaces, International Mathematics Research Notices, Volume 2021, Issue 17, 12993–13010.
  • S. Arthamonov, N. Reshetikhin, Superintegrable Systems on Moduli Spaces of Flat Connections, Commun. Math. Phys. 386, 1337-1381 (2021)

Geometry related to superintegrable systems:

  • R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), 375422,
  • Kiesenhofer, A., Miranda, E., Non-commutative integrable systems on b-symplectic manifolds, Regul. Chaotic Dyn. 21 (2016), no. 6, 643659.

Quantum spin Calogero-Moser systems and their superintegrability:

  • K. Hikami, M. Wadati, Integrability of Calogero-Moser spin system, Journal of the Physics Society of Japan, v. 62, n. 2 (1993) 469-472.
  • Kuznetsov, Vadim B., Hidden symmetry of the quantum Calogero-Moser sys- tem. Phys. Lett. A 218 (1996), no. 3-6, 212222.
  • Feher, L.; Pusztai, B. G., Twisted spin Sutherland models from quantum Hamiltonian reduction. J. Phys. A 41 (2008), no. 19, 194009.
  • G. Schrader and A. Shapiro Continuous tensor categories from quantum groups I: algebraic aspects. arXiv preprint arXiv:1708.08107 (2017)
  • E. Koelink, M. van Pruijssen, P. Roman, Matrix elements of irreducible representations of SU(n+1)X SU(n+1) and multivariable matrix-valued orthogonal polynomials. ArXiv: 1706.01927
  • J. Stokman, N. Reshetikhin, N-point spherical functions and asymptotic boundary KZB equations, arXiv: 2002.202251.
  • N. Reshetikhin, J. Stokman, Asymptotic boundary KZB operators and quantum Calogero-Moser spin chain, arXiv: 2012.13497.