June  2021, 41(6): 2559-2599. doi: 10.3934/dcds.2020375

Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems

1. 

Department of Mathematics, Polytechnic university of Catalonia, C. Pau Gargallo, 14, 08028 Barcelona, Spain

2. 

Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Belgrade, Serbia

* Corresponding author: Božidar Jovanović

Received  February 2020 Revised  September 2020 Published  June 2021 Early access  November 2020

We study geometric and algebraic geometric properties of the continuous and discrete Neumann systems on cotangent bundles of Stiefel varieties $ V_{n,r} $. The systems are integrable in the non-commutative sense, and by applying a $ 2r\times 2r $–Lax representation, we show that generic complex invariant manifolds are open subsets of affine Prym varieties on which the complex flow is linear. The characteristics of the varieties and the direction of the flow are calculated explicitly. Next, we construct a family of multi-valued integrable discretizations of the Neumann systems and describe them as translations on the Prym varieties, which are written explicitly in terms of divisors of points on the spectral curve.

Citation: Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2559-2599. doi: 10.3934/dcds.2020375
References:
[1]

M. R. AdamsJ. Harnad and E. Previato, Isospectral Hamiltonian flows in finite and infinite dimensions. Ⅰ. Generalized Moser systems and moment maps into loop algebras, Comm. Math. Phys., 117 (1988), 451-500.  doi: 10.1007/BF01223376.

[2]

M. R. AdamsJ. Harnad and J. Hurtubise, Isospectral Hamiltonian flows in finite and infinite dimensions Ⅱ. Integration of flows, Comm. Math. Phys., 134 (1990), 555-585.  doi: 10.1007/BF02098447.

[3]

M. R. AdamsJ. Harnad and J. Hurtubise, Dual moment maps into loop algebras, Lett. Math. Phys., 20 (1990), 299-308.  doi: 10.1007/BF00626526.

[4]

M. R. AdamsJ. Harnad and J. Hurtubise, Darboux coordinates on coadjoint orbits of Lie algebras, Lett. Math. Phys., 40 (1997), 41-57.  doi: 10.1023/A:1007355508426.

[5]

M. Adler and P. van Moerbeke, Birkhoff strata, Bäckund transformations and regularization of isospectral operators, Advances in Math., 108 (1994), 140-204.  doi: 10.1006/aima.1994.1070.

[6]

M. Adler, P. van Moerbeke and P. Vanheacke, Algebraic Integrability, Painleve Geometry and Lie Algebras, Springer, 2004. doi: 10.1007/978-3-662-05650-9.

[7]

C. Athorne and A. Fordy, Generalized KdV and mKdV equations associated with symmetric spaces, J. Phys.A., 20 (1987), 1377-1386.  doi: 10.1088/0305-4470/20/6/021.

[8]

E. D. Belokolos, A. I. Bobenko, V. Z. Enol'sii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer Series in Nonlinear Dynamics. Springer–Verlag, 1994.

[9]

A. Beauville, Prym varieties and the Schottky problem, Invent. Math., 41 (1977), 149-196.  doi: 10.1007/BF01418373.

[10]

A. Beauville, Jacobiennes des courbes spectrales et syst'emes hamiltoniens completement integrables, Acta Math., 164 (1990), 211-235.  doi: 10.1007/BF02392754.

[11]

O. I. Bogoyavlenski, New integrable problem of classical mechanics, Comm. Math. Phys., 94 (1984), 255-269.  doi: 10.1007/BF01209304.

[12]

A. V. Bolsinov and B. Jovanović, Noncommutative integrability, moment map and geodesic flows, Annals of Global Analysis and Geometry, 23 (2003), 305–322, arXiv: math-ph/0109031. doi: 10.1023/A:1023023300665.

[13]

E. Casas-Alvero, Singularities of Plane Curves, London Math. Soc. Lecture Notes Series. 276 Cambridge University Press, 2000. doi: 10.1017/CBO9780511569326.

[14]

A. Clebsch and P. Gordan, Theorie Der Abelschen Funktionen, Teubner, Leipzig, 1866.

[15]

J. EilbeckV. Enol'skiV. Kuznetzov and A. Tsiganov, Linear $mathbb{R}^{N}$-matrix algebra for classical separable systems, J. Phys. A: Math. Gen., 27 (1994), 567-578.  doi: 10.1088/0305-4470/27/2/038.

[16]

P. A. Dirac, On generalized Hamiltonian dynamics, Can. J. Math., 2 (1950), 129-148.  doi: 10.4153/CJM-1950-012-1.

[17]

V. Dragović and B. Gajić, The Lagrange bitop on $so(4) \times so(4)$ and geometry of the Prym varieties, American J. of Math., 126 (2004), 981–1004, arXiv: math-ph/0201036. doi: 10.1353/ajm.2004.0035.

[18]

B. A. Dubrovin, Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties, Funct.Anal.Appl., 11 (1977), 265-277. 

[19]

B. A Dubrovin, S. P. Novikov and I. M. Krichever, Integrable Systems. I,, in Itogi Nauki i Tekhniki. Sovr.Probl.Mat. Fund.Naprav., 4 (1985), 179–284. English transl.: Encyclopaedia of Math.Sciences, Vol. 4, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-06791-8_3.

[20]

J. Fay, Theta-functions on Riemann Surfaces, , Springer Lecture Notes, 352, Springer-Verlag, 1973.

[21]

Yu. Fedorov, Classical integrable systems related to generalized Jacobians, Acta Appl. Math., 55 (1999), 251-301.  doi: 10.1023/A:1006178224117.

[22]

Yu. Fedorov, Bäcklund transformations on coadjoint orbits of the loop algebra $\widetilde {{\rm{gl}}}({r})$, Recent advances in integrable systems (Kowloon, Hong Kong, 2000). J. Nonlinear Math. Phys, 9 (2002), suppl. 1, 29–46. doi: 10.2991/jnmp.2002.9.s1.3.

[23]

Yu. Fedorov, Integrable flows and Bäcklund transformations on extended Stiefel varieies with application to the Euler top on the Lie group $SO(3)$, J. Non. Math. Phys., 12 (2005), Suppl. 2, 77–94, arXiv: nlin/0505045. doi: 10.2991/jnmp.2005.12.s2.7.

[24]

Yu. Fedorov and B. Jovanović, Geodesic flows and neumann systems on stiefel varieties: Geometry and integrability, Math. Z., 270 (2012), 659–698, arXiv: 1011.1835. doi: 10.1007/s00209-010-0818-y.

[25]

Yu Fedorov and B. Jovanović, Three natural mechanical systems on Stiefel varieties, J. Phys. A., 45 (2012), 165204, (13pp), arXiv: 1202.1660. doi: 10.1088/1751-8113/45/16/165204.

[26]

R. L. Fernandes and P. Vanhaecke, Hyperelliptic Prym varieties and integrable systems, Comm. Math. Phys., 221 (2001), 169–196, arXiv: math-ph/0011051. doi: 10.1007/s002200100476.

[27]

L. Gavrilov, Generalized Jacobians of spectral curves and completely integrable systems, Math. Z., 230 (1999), 487-508.  doi: 10.1007/PL00004701.

[28]

L. Gavrilov, Jacobians of singularized spectral curves and completely integrable systems, in The Kowalevski Property (Leeds, 2000), 59–68, CRM Proc. Lecture Notes, 32, Amer. Math. Soc., Providence, RI, 2002, arXiv: math/0111235.

[29]

G. Jensen, Einstein metrics on principal fiber bundles, J. Diff. Geom., 8 (1973), 599-614.  doi: 10.4310/jdg/1214431962.

[30]

B. Jovanović and Yu. Fedorov, Discrete geodesic flows on Stiefel manifolds, Tr. Mat. Inst. Steklova, 310 (2020) 176-–188, (Russian); English transl.: Proceedings of the Steklov Institute of Mathematics, 310 (2020), 163–-174. doi: 10.4213/tm4107.

[31]

B. Jovanović and V. Jovanović, Virtual billiards in pseudo–Euclidean spaces: Discrete Hamiltonian and contact integrability, Discrete and Continuous Dynamical Systems–Series A, 37 (2017), 5163–5190, arXiv: 1510.04037. doi: 10.3934/dcds.2017224.

[32]

B. Jovanović and V. Jovanović, Heisenberg model in pseudo-Euclidean spaces Ⅱ, Regular and Chaotic Dynamics, 23 (2018), 418–437, arXiv: 1808.10783. doi: 10.1134/S1560354718040044.

[33]

A. N. Hone, V. B. Kuznetsov and O. Ragnisco, Bäcklund transformations for many-body systems related to KdV, J. Phys. A, 32 (1999), L299–L306, arXiv: solv-int/9904003. doi: 10.1088/0305-4470/32/27/102.

[34]

R. InoueY. Konishi and T. Yamazaki, Jacobian variety and integrable system–-after Mumford, Beauville and Vanhaecke, J. Geom. Phys., 57 (2007), 815-831.  doi: 10.1016/j.geomphys.2006.06.004.

[35]

S. Kapustin, The Neumann system on Stiefel varieties, Preprint, 1992 (Russian).

[36]

F. Kirwan, Complex Algebraic Curves, London Mathematical Society Student Texts, 23. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511623929.

[37]

H. Knörrer, Geodesics on quadrics and a mechanical problem of C. Neumann, J. Reine Angew. Math., 334 (1982), 69-78. 

[38]

I. Krichiver, Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv., 32 (1977), 185-213. 

[39]

V. Kuznetsov and P. Vanhaecke, Bäcklund transformations for finite-dimensional integrable systems: A geometric approach, J. Geom. Phys., 44 (2002), 1–40, arXiv: nlin/0004003. doi: 10.1016/S0393-0440(02)00029-3.

[40]

H. P. McKean, Variation on a theme of Jacobi, Comm. Pure Appl. Math., 38 (1985), 669-678.  doi: 10.1002/cpa.3160380514.

[41]

A. S. Mishchenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121. 

[42]

P. van Moerbeke and D. Mumford, The spectrum of difference operators and algebraic curves, Acta Math., 143 (1979), 93-154.  doi: 10.1007/BF02392090.

[43]

J. Moser, Geometry of quadric and spectral theory, in: Chern Symposium 1979, Berlin–Heidelberg–New York, 1980,147–188.

[44]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243.  doi: 10.1007/BF02352494.

[45]

D. Mumford, Tata Lectures on Theta II, Progress in Math., Birkhauser, 1984.

[46]

N. N. Nekhoroshev, Action-angle variables and their generalization, Trans. Mosc. Math. Soc., 26 (1972), 180-198. 

[47]

C. Neumann, De probleme quodam mechanico, quod ad primam integralium ultra-ellipticoram classem revocatum, J. Reine Angew. Math., 56 (1859), 46-63.  doi: 10.1515/crll.1859.56.46.

[48]

A. M. Perelomov, Some remarks on the integrability of the equations of motion of a rigid body in an ideal fluid, Funct. Anal. Appl., 15 (1981), 144-146. 

[49]

M. Pedroni and P. Vanhaecke, A Lie algebraic generalization of the Mumford system, its symmetries and its multi-Hamiltonian structure, Regul. Chaotic Dyn., 3 (1998), 132-160.  doi: 10.1070/rd1998v003n03ABEH000086.

[50]

J. Potter, Matrix quadratic solutions, J. SIAM Appl. Math., 14 (1966), 496-501.  doi: 10.1137/0114044.

[51]

E. Previato, Flows on $r$-gonal Jacobians,, in: The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985), Contemp. Math., 64, Amer. Math. Soc., Providence, RI, 1987,153–180. doi: 10.1090/conm/064/881461.

[52]

O. Ragnisco, A discrete Neumann system, Phys.Lett.A., 167 (1992), 165-171.  doi: 10.1016/0375-9601(92)90222-8.

[53]

A. G. Reyman and M. A. Semonov-Tian-Shanski, Group theoretical methods in the theory of finite dimensional integrable systems,, in: Integrable Systems. VII, Itogi Nauki i Tekhniki. Sovr.Probl.Mat. Fund.Naprav, 16, VINITI, Moscow 1987,116–225 (Russian). English transl.: Encyclopaedia of Math.Sciences, 16, Dynamical systems VII, Springer 1994. https://www.springer.com/gp/book/9783540181767. doi: 10.1007/978-3-662-06796-3_7.

[54]

J. P. Serre, Groupes Algébriques et Corps De Classes, Hermann, Paris, 1959.

[55]

R. J. Schilling, Generalizations of the Neumann system. A curve theoretical approach. II, Comm. Pure Appl. Math., 42 (1989), 409-442.  doi: 10.1002/cpa.3160420404.

[56]

Yu. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach, Progress in Mathematics, 219. Birkhauser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-8016-9.

[57]

P. Vanhaecke, Integrable Systems in the Realm of Algebraic Geometry, Springer Lecture Notes. 1638, 1996. doi: 10.1007/978-3-662-21535-7.

[58]

A. P. Veselov, Integrable discrete-time systems and difference operators, Funct. An. and Appl., 22 (1988), 83-93.  doi: 10.1007/BF01077598.

[59]

A. P. Veselov, Integrable maps, Russ. Math. Surv., 46 (1991), 1-51.  doi: 10.1070/RM1991v046n05ABEH002856.

[60]

O. Vivolo, Jacobians of singular spectral curves and completely integrable systems, Proc. Edinburg Math. Soc., 43 (2000), 605-623.  doi: 10.1017/S0013091500021222.

show all references

References:
[1]

M. R. AdamsJ. Harnad and E. Previato, Isospectral Hamiltonian flows in finite and infinite dimensions. Ⅰ. Generalized Moser systems and moment maps into loop algebras, Comm. Math. Phys., 117 (1988), 451-500.  doi: 10.1007/BF01223376.

[2]

M. R. AdamsJ. Harnad and J. Hurtubise, Isospectral Hamiltonian flows in finite and infinite dimensions Ⅱ. Integration of flows, Comm. Math. Phys., 134 (1990), 555-585.  doi: 10.1007/BF02098447.

[3]

M. R. AdamsJ. Harnad and J. Hurtubise, Dual moment maps into loop algebras, Lett. Math. Phys., 20 (1990), 299-308.  doi: 10.1007/BF00626526.

[4]

M. R. AdamsJ. Harnad and J. Hurtubise, Darboux coordinates on coadjoint orbits of Lie algebras, Lett. Math. Phys., 40 (1997), 41-57.  doi: 10.1023/A:1007355508426.

[5]

M. Adler and P. van Moerbeke, Birkhoff strata, Bäckund transformations and regularization of isospectral operators, Advances in Math., 108 (1994), 140-204.  doi: 10.1006/aima.1994.1070.

[6]

M. Adler, P. van Moerbeke and P. Vanheacke, Algebraic Integrability, Painleve Geometry and Lie Algebras, Springer, 2004. doi: 10.1007/978-3-662-05650-9.

[7]

C. Athorne and A. Fordy, Generalized KdV and mKdV equations associated with symmetric spaces, J. Phys.A., 20 (1987), 1377-1386.  doi: 10.1088/0305-4470/20/6/021.

[8]

E. D. Belokolos, A. I. Bobenko, V. Z. Enol'sii, A. R. Its and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer Series in Nonlinear Dynamics. Springer–Verlag, 1994.

[9]

A. Beauville, Prym varieties and the Schottky problem, Invent. Math., 41 (1977), 149-196.  doi: 10.1007/BF01418373.

[10]

A. Beauville, Jacobiennes des courbes spectrales et syst'emes hamiltoniens completement integrables, Acta Math., 164 (1990), 211-235.  doi: 10.1007/BF02392754.

[11]

O. I. Bogoyavlenski, New integrable problem of classical mechanics, Comm. Math. Phys., 94 (1984), 255-269.  doi: 10.1007/BF01209304.

[12]

A. V. Bolsinov and B. Jovanović, Noncommutative integrability, moment map and geodesic flows, Annals of Global Analysis and Geometry, 23 (2003), 305–322, arXiv: math-ph/0109031. doi: 10.1023/A:1023023300665.

[13]

E. Casas-Alvero, Singularities of Plane Curves, London Math. Soc. Lecture Notes Series. 276 Cambridge University Press, 2000. doi: 10.1017/CBO9780511569326.

[14]

A. Clebsch and P. Gordan, Theorie Der Abelschen Funktionen, Teubner, Leipzig, 1866.

[15]

J. EilbeckV. Enol'skiV. Kuznetzov and A. Tsiganov, Linear $mathbb{R}^{N}$-matrix algebra for classical separable systems, J. Phys. A: Math. Gen., 27 (1994), 567-578.  doi: 10.1088/0305-4470/27/2/038.

[16]

P. A. Dirac, On generalized Hamiltonian dynamics, Can. J. Math., 2 (1950), 129-148.  doi: 10.4153/CJM-1950-012-1.

[17]

V. Dragović and B. Gajić, The Lagrange bitop on $so(4) \times so(4)$ and geometry of the Prym varieties, American J. of Math., 126 (2004), 981–1004, arXiv: math-ph/0201036. doi: 10.1353/ajm.2004.0035.

[18]

B. A. Dubrovin, Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties, Funct.Anal.Appl., 11 (1977), 265-277. 

[19]

B. A Dubrovin, S. P. Novikov and I. M. Krichever, Integrable Systems. I,, in Itogi Nauki i Tekhniki. Sovr.Probl.Mat. Fund.Naprav., 4 (1985), 179–284. English transl.: Encyclopaedia of Math.Sciences, Vol. 4, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-06791-8_3.

[20]

J. Fay, Theta-functions on Riemann Surfaces, , Springer Lecture Notes, 352, Springer-Verlag, 1973.

[21]

Yu. Fedorov, Classical integrable systems related to generalized Jacobians, Acta Appl. Math., 55 (1999), 251-301.  doi: 10.1023/A:1006178224117.

[22]

Yu. Fedorov, Bäcklund transformations on coadjoint orbits of the loop algebra $\widetilde {{\rm{gl}}}({r})$, Recent advances in integrable systems (Kowloon, Hong Kong, 2000). J. Nonlinear Math. Phys, 9 (2002), suppl. 1, 29–46. doi: 10.2991/jnmp.2002.9.s1.3.

[23]

Yu. Fedorov, Integrable flows and Bäcklund transformations on extended Stiefel varieies with application to the Euler top on the Lie group $SO(3)$, J. Non. Math. Phys., 12 (2005), Suppl. 2, 77–94, arXiv: nlin/0505045. doi: 10.2991/jnmp.2005.12.s2.7.

[24]

Yu. Fedorov and B. Jovanović, Geodesic flows and neumann systems on stiefel varieties: Geometry and integrability, Math. Z., 270 (2012), 659–698, arXiv: 1011.1835. doi: 10.1007/s00209-010-0818-y.

[25]

Yu Fedorov and B. Jovanović, Three natural mechanical systems on Stiefel varieties, J. Phys. A., 45 (2012), 165204, (13pp), arXiv: 1202.1660. doi: 10.1088/1751-8113/45/16/165204.

[26]

R. L. Fernandes and P. Vanhaecke, Hyperelliptic Prym varieties and integrable systems, Comm. Math. Phys., 221 (2001), 169–196, arXiv: math-ph/0011051. doi: 10.1007/s002200100476.

[27]

L. Gavrilov, Generalized Jacobians of spectral curves and completely integrable systems, Math. Z., 230 (1999), 487-508.  doi: 10.1007/PL00004701.

[28]

L. Gavrilov, Jacobians of singularized spectral curves and completely integrable systems, in The Kowalevski Property (Leeds, 2000), 59–68, CRM Proc. Lecture Notes, 32, Amer. Math. Soc., Providence, RI, 2002, arXiv: math/0111235.

[29]

G. Jensen, Einstein metrics on principal fiber bundles, J. Diff. Geom., 8 (1973), 599-614.  doi: 10.4310/jdg/1214431962.

[30]

B. Jovanović and Yu. Fedorov, Discrete geodesic flows on Stiefel manifolds, Tr. Mat. Inst. Steklova, 310 (2020) 176-–188, (Russian); English transl.: Proceedings of the Steklov Institute of Mathematics, 310 (2020), 163–-174. doi: 10.4213/tm4107.

[31]

B. Jovanović and V. Jovanović, Virtual billiards in pseudo–Euclidean spaces: Discrete Hamiltonian and contact integrability, Discrete and Continuous Dynamical Systems–Series A, 37 (2017), 5163–5190, arXiv: 1510.04037. doi: 10.3934/dcds.2017224.

[32]

B. Jovanović and V. Jovanović, Heisenberg model in pseudo-Euclidean spaces Ⅱ, Regular and Chaotic Dynamics, 23 (2018), 418–437, arXiv: 1808.10783. doi: 10.1134/S1560354718040044.

[33]

A. N. Hone, V. B. Kuznetsov and O. Ragnisco, Bäcklund transformations for many-body systems related to KdV, J. Phys. A, 32 (1999), L299–L306, arXiv: solv-int/9904003. doi: 10.1088/0305-4470/32/27/102.

[34]

R. InoueY. Konishi and T. Yamazaki, Jacobian variety and integrable system–-after Mumford, Beauville and Vanhaecke, J. Geom. Phys., 57 (2007), 815-831.  doi: 10.1016/j.geomphys.2006.06.004.

[35]

S. Kapustin, The Neumann system on Stiefel varieties, Preprint, 1992 (Russian).

[36]

F. Kirwan, Complex Algebraic Curves, London Mathematical Society Student Texts, 23. Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511623929.

[37]

H. Knörrer, Geodesics on quadrics and a mechanical problem of C. Neumann, J. Reine Angew. Math., 334 (1982), 69-78. 

[38]

I. Krichiver, Methods of algebraic geometry in the theory of non-linear equations, Russ. Math. Surv., 32 (1977), 185-213. 

[39]

V. Kuznetsov and P. Vanhaecke, Bäcklund transformations for finite-dimensional integrable systems: A geometric approach, J. Geom. Phys., 44 (2002), 1–40, arXiv: nlin/0004003. doi: 10.1016/S0393-0440(02)00029-3.

[40]

H. P. McKean, Variation on a theme of Jacobi, Comm. Pure Appl. Math., 38 (1985), 669-678.  doi: 10.1002/cpa.3160380514.

[41]

A. S. Mishchenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121. 

[42]

P. van Moerbeke and D. Mumford, The spectrum of difference operators and algebraic curves, Acta Math., 143 (1979), 93-154.  doi: 10.1007/BF02392090.

[43]

J. Moser, Geometry of quadric and spectral theory, in: Chern Symposium 1979, Berlin–Heidelberg–New York, 1980,147–188.

[44]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243.  doi: 10.1007/BF02352494.

[45]

D. Mumford, Tata Lectures on Theta II, Progress in Math., Birkhauser, 1984.

[46]

N. N. Nekhoroshev, Action-angle variables and their generalization, Trans. Mosc. Math. Soc., 26 (1972), 180-198. 

[47]

C. Neumann, De probleme quodam mechanico, quod ad primam integralium ultra-ellipticoram classem revocatum, J. Reine Angew. Math., 56 (1859), 46-63.  doi: 10.1515/crll.1859.56.46.

[48]

A. M. Perelomov, Some remarks on the integrability of the equations of motion of a rigid body in an ideal fluid, Funct. Anal. Appl., 15 (1981), 144-146. 

[49]

M. Pedroni and P. Vanhaecke, A Lie algebraic generalization of the Mumford system, its symmetries and its multi-Hamiltonian structure, Regul. Chaotic Dyn., 3 (1998), 132-160.  doi: 10.1070/rd1998v003n03ABEH000086.

[50]

J. Potter, Matrix quadratic solutions, J. SIAM Appl. Math., 14 (1966), 496-501.  doi: 10.1137/0114044.

[51]

E. Previato, Flows on $r$-gonal Jacobians,, in: The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985), Contemp. Math., 64, Amer. Math. Soc., Providence, RI, 1987,153–180. doi: 10.1090/conm/064/881461.

[52]

O. Ragnisco, A discrete Neumann system, Phys.Lett.A., 167 (1992), 165-171.  doi: 10.1016/0375-9601(92)90222-8.

[53]

A. G. Reyman and M. A. Semonov-Tian-Shanski, Group theoretical methods in the theory of finite dimensional integrable systems,, in: Integrable Systems. VII, Itogi Nauki i Tekhniki. Sovr.Probl.Mat. Fund.Naprav, 16, VINITI, Moscow 1987,116–225 (Russian). English transl.: Encyclopaedia of Math.Sciences, 16, Dynamical systems VII, Springer 1994. https://www.springer.com/gp/book/9783540181767. doi: 10.1007/978-3-662-06796-3_7.

[54]

J. P. Serre, Groupes Algébriques et Corps De Classes, Hermann, Paris, 1959.

[55]

R. J. Schilling, Generalizations of the Neumann system. A curve theoretical approach. II, Comm. Pure Appl. Math., 42 (1989), 409-442.  doi: 10.1002/cpa.3160420404.

[56]

Yu. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach, Progress in Mathematics, 219. Birkhauser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-8016-9.

[57]

P. Vanhaecke, Integrable Systems in the Realm of Algebraic Geometry, Springer Lecture Notes. 1638, 1996. doi: 10.1007/978-3-662-21535-7.

[58]

A. P. Veselov, Integrable discrete-time systems and difference operators, Funct. An. and Appl., 22 (1988), 83-93.  doi: 10.1007/BF01077598.

[59]

A. P. Veselov, Integrable maps, Russ. Math. Surv., 46 (1991), 1-51.  doi: 10.1070/RM1991v046n05ABEH002856.

[60]

O. Vivolo, Jacobians of singular spectral curves and completely integrable systems, Proc. Edinburg Math. Soc., 43 (2000), 605-623.  doi: 10.1017/S0013091500021222.

[1]

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