The Lax–Sato integrable heavenly equations on functional supermanifolds and their Lie-algebraic structure


A Lie-algebraic approach to constructing the Lax–Sato integrable superanalogs of heavenly equations by use of the loop Lie algebra of superconformal vector fields on a 1|N-dimensional supertorus is proposed. In the framework of this approach integrable superanalogs of the Mikhalev–Pavlov heavenly equation are obtained for all \(N\in {\mathbb {N}} {\setminus } \{ 4,5 \}\) as well as Shabat type reductions for all \(N\in {\mathbb {N}}\). The Lax–Sato integrable superanalogs of the generalized Liouville heavenly equations are found by means of the Lie algebra of holomorphic in “spectral” parameter superconformal vector fields on a 1|N-dimensional complex supertorus.

This is a preview of subscription content, log in to check access.


  1. 1.

    Berezin, F.A.: Introduction to Algebra and Analysis with Anticommuting Variables. Moscow State University, Moscow (1983) (in Russian)

  2. 2.

    Blackmore, D., Hentosh, E.O., Prykarpatski, A.K.: The novel Lie-algebraic approach to studying integrable heavenly type multi-dimensional dynamical systems. J. Gen. Lie Theory Appl. 11(3), # 286 (2017)

    Google Scholar 

  3. 3.

    Blackmore, D., Prykarpatsky, A.K., Samoylenko, V.Hr.: Nonlinear Dynamical Systems of Mathematical Physics. World Scientific, Hackensack (2011)

  4. 4.

    Bogdanov, L.V., Dryuma, V.S., Manakov, S.V.: Dunajski generalization of the second heavenly equation: dressing method and the hierarchy. J. Phys. A 40(48), 14383–14393 (2007)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bogdanov, L.V., Konopelchenko, B.G.: Grassmannians \({\rm Gr}(N-1, N+1)\), closed differential \(N-1\)-forms and \(N\)-dimensional integrable systems. J. Phys. A 46(8), # 085201 (2013). arXiv:1208.6129v2

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bogdanov, L.V., Pavlov, M.V.: Linearly degenerate hierarchies of quasiclassical SDYM type. J. Math. Phys. 58(9), # 093505 (2017). arXiv:1603.00238v2

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dunajski, M., Ferapontov, E.V., Kruglikov, B.: On the Einstein–Weyl and conformal self-duality equations. J. Math. Phys. 56(8), #083501 (2015)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dunajski, M., Kryński, W.: Einstein-Weyl geometry, dispersionless Hirota equation and Veronese webs. Math. Proc. Cambridge Philos. Soc. 157(1), 139–150 (2014). arXiv:1301.0621v2

    MathSciNet  Article  Google Scholar 

  9. 9.

    Faddeev, L.D., Takhtadjan, L.A.: Hamiltonian Methods in the Theory of Solitons. Classics in Mathematics. Springer, Berlin (2007)

    Google Scholar 

  10. 10.

    Ferapontov, E.V., Moss, J.: Linearly degenerate partial differential equations and quadratic line complexes. Comm. Anal. Geom. 23(1), 91–127 (2015). arXiv:1204.2777v1

    MathSciNet  Article  Google Scholar 

  11. 11.

    Hentosh, O.Ye.: Compatibly bi-Hamiltonian superconformal analogs of Lax-integrable nonlinear dynamical systems. Ukrainian Math. J. 58(7), 1001–1015 (2006)

  12. 12.

    Hentosh, O.E.: The Lax integrable Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems and its finite-dimensional Neumann type reduction. Ukrainian Math. J. 61(7), 1075–1092 (2009)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Hentosh, O.E., Prykarpatsky, Ya.A., Blackmore, D., Prykarpatski, A.K.: Lie-algebraic structure of Lax–Sato integrable heavenly equations and the Lagrange–d’Alembert principle. J. Geom. Phys. 120, 208–227 (2017)

  14. 14.

    Kruglikov, B., Morozov, O.: Integrable dispersionless PDEs in 4D, their symmetry pseudogroups and deformations. Lett. Math. Phys. 105(12), 1703–1723 (2015)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Kulish, P.P.: Analog of the Korteweg–de Vries equation for the superconformal algebra. J. Soviet Math. 41(2), 970–975 (1988)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Manakov, S.V., Santini, P.M.: Inverse scattering problem for vector fields and the Cauchy problem for the heavenly equation. Phys. Lett. A 359(6), 613–619 (2006)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Martínez Alonso, L.M., Shabat, A.B.: Hydrodynamic reductions and solutions of a universal hierarchy. Theor. Math. Phys. 140(2), 1073–1085 (2004)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Mikhalev, V.G.: On the Hamiltonian formalism of Korteweg–de Vries type hierarchies. Funct. Anal. Appl. 26(2), 140–142 (1992)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Misiołek, G.: A shallow water equation as a geodesic flow on the Bott–Virasoro group. J. Geom. Phys. 24(3), 203–208 (1998)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Ovsienko, V.: Bi-Hamiltonian nature of the equation \(u_{tx}=u_{xy}u_{y}-u_{yy}u_{x}\). Adv. Pure Appl. Math. 1(1), 7–17 (2010). arXiv:0802.1818v1

    MathSciNet  Article  Google Scholar 

  21. 21.

    Ovsienko, V., Roger, C.: Looped cotangent Virasoro algebra and non-linear integrable systems in dimension \(2+1\). Comm. Math. Phys. 273(2), 357–378 (2007)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Pavlov, M.V.: Integrable hydrodynamic chains. J. Math. Phys. 44(9), 4134–4156 (2003)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Plebański, J.F.: Some solutions of complex Einstein equations. J. Math. Phys. 16(12), 2395–2402 (1975)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Prykarpatski, A.K., Hentosh, O.E., Prykarpatsky, Ya.A.: Geometric structure of the classical Lagrange–d’Alambert principle and its application to the integrable nonlinear dynamical systems. Mathematics 5(4), # 75 (2017)

  25. 25.

    Radul, A.O.: Lie algebras of differential operators, their central extensions and \(W\)-algebras. Funct. Anal. Appl. 25(1), 25–39 (1991)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Reyman, A.G., Semenov-Tian-Shansky, M.A.: Integrable Systems. Computer Research Institute, Moscow-Izhevsk (2003) (in Russian)

  27. 27.

    Sergyeyev, A., Szablikowski, B.M.: Central extensions of cotangent universal hierarchy: \((2+1)\)-dimensional bi-Hamiltonian systems. Phys. Lett. A 372(47), 7016–7023 (2008)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Sheftel, M.B., Yazıcı, D., Malykh, A.A.: Recursion operators and bi-Hamiltonian structure of the general heavenly equation. J. Geom. Phys. 116, 124–139 (2017)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Takasaki, K., Takebe, T.: Integrable hierarchies and dispersionless limit. Rev. Math. Phys. 7(5), 743–808 (1995)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Vladimirov, V.S., Volovich, I.V.: Superanalysis. I. Differential calculus. Theor. Math. Phys. 59(1), 317–335 (1984)

    MathSciNet  Article  Google Scholar 

Download references


The authors cordially thank Prof. Maciej Błaszak, Prof. Błażej Szablikowski and Prof. Jan Cieśliński for fruitful discussions during the International Conference in Functional Analysis dedicated to the 125th anniversary of Stefan Banach held on September 18–23, 2017, in Lviv, Ukraine.

Author information



Corresponding author

Correspondence to Yarema Prykarpatsky.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hentosh, O., Prykarpatsky, Y. The Lax–Sato integrable heavenly equations on functional supermanifolds and their Lie-algebraic structure. European Journal of Mathematics 6, 232–247 (2020).

Download citation


  • Heavenly type equations
  • Lax–Sato integrability
  • Superconformal vector fields
  • Adler–Kostant–Symes theory
  • Casimir invariants

Mathematics Subject Classification

  • 37K05
  • 37K30
  • 37K65
  • 35Q75
  • 35Q35
  • 17B80
  • 58C50