@article{Hentosh_Prykarpatskyy_Balinsky_Prykarpatski_2022, title={Geometric structures on the orbits of loop diffeomorphism groups and related “heavenly-type” Hamiltonian systems. I}, volume={74}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/6614}, DOI={10.37863/umzh.v74i8.6614}, abstractNote={<p>UDC 517.9</p> <p><span class="Apple-converted-space"> </span>review of differential-geometric and Lie-algebraic approaches to the study of a broad class of nonlinear integrable <span class="Apple-converted-space"> </span>differential systems of ``heavenly’’ type associated with Hamiltonian flows on the spaces conjugate to the loop Lie algebras of vector fields on the tori.<span class="Apple-converted-space"> </span>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.<span class="Apple-converted-space"> </span>The corresponding hierarchies of conservation laws and their relationships with Casimir invariants are analyzed.<span class="Apple-converted-space"> </span>Typical examples of these systems are considered and their complete integrability is established by using the developed Lie-algebraic construction.<span class="Apple-converted-space"> </span>We describe new generalizations of the integrable dispersion-free systems of ``heavenly’’ type for which the corresponding generating elements of orbits have a factorized structure, which allows their extension to the multidimensional case.</p>}, number={8}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Hentosh , O. E. and Prykarpatskyy , Ya. A. and Balinsky , A. A. and Prykarpatski , A. K.}, year={2022}, month={Oct.}, pages={1029 - 1059} }