{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:34:05Z","timestamp":1760142845535,"version":"build-2065373602"},"reference-count":64,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,5]],"date-time":"2024-01-05T00:00:00Z","timestamp":1704412800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Department of Mathematical Sciences at the UAEU","award":["G00003658","G00004159"],"award-info":[{"award-number":["G00003658","G00004159"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids and their related R structures generated by suitably defined tensor mappings. Lie\u2013Poisson brackets that are invariant with respect to the coadjoint action of the loop diffeomorphism group are created, and the related Courant-type algebroids are described. The corresponding integrable Hamiltonian flows generated by Casimir functionals and generalizing so-called heavenly-type differential systems describing diverse geometric structures of conformal type in finite dimensional Riemannian manifolds are described.<\/jats:p>","DOI":"10.3390\/sym16010076","type":"journal-article","created":{"date-parts":[[2024,1,5]],"date-time":"2024-01-05T10:06:43Z","timestamp":1704449203000},"page":"76","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Some Aspects of the Courant-Type Algebroids, the Related Coadjoint Orbits and Integrable Systems"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5124-5890","authenticated-orcid":false,"given":"Anatolij K.","family":"Prykarpatski","sequence":"first","affiliation":[{"name":"Department of Computer Science and Telecommunication, Cracow University of Technology, 31-155 Krakow, Poland"},{"name":"Department of Advanced Mathematics, Lviv Polytechnic National University, 79000 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5750-163X","authenticated-orcid":false,"given":"Victor A.","family":"Bovdi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, United Arab Emirates University, Al Ain P.O. Box 15551, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1112\/blms\/27.2.97","article-title":"Lie algebroids and Lie pseudoalgebras","volume":"27","author":"Mackenzie","year":"1995","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Mackenzie, K. (1987). Lie Groupoids and Lie Algebroids in Differential Geometry, Cambridge University Press. MR 89g:58225.","DOI":"10.1017\/CBO9780511661839"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1353\/ajm.2001.0017","article-title":"Leibniz Algebras, Courant Algebroids, and Multiplications on Reductive Homogeneous Spaces","volume":"123","author":"Kinyon","year":"2000","journal-title":"Am. 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