We present a review of differential-geometric and Lie-algebraic approaches to the study of a broad class of nonlinear integrable differential systems of “heavenly” type associated with Hamiltonian flows on the spaces conjugated to the loop Lie algebras of vector fields on the tori. These flows are generated by the corresponding orbits of the coadjoint action of the diffeomorphism loop group and satisfy the Lax–Sato-type vector-field compatibility conditions. The corresponding hierarchies of conservation laws and their relationships with Casimir invariants are analyzed. We consider typical examples of these systems and establish their complete integrability by using the developed Lie-algebraic construction. We also describe new generalizations of the integrable dispersion-free systems of heavenly type for which the corresponding generating elements of the orbits have factorized structures, which allows their extension to the multidimensional case.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 9, pp. 1182–1200, September, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i9.7234.
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Hentosh, O.E., Prykarpatskyy, Y., Balinsky, A.A. et al. Geometric Structures on the Orbits of Loop Diffeomorphism Groups and Related Heavenly-Type Hamiltonian Systems. II. Ukr Math J 74, 1348–1368 (2023). https://doi.org/10.1007/s11253-023-02140-7
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DOI: https://doi.org/10.1007/s11253-023-02140-7