{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T06:35:17Z","timestamp":1764570917112,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,10,10]],"date-time":"2020-10-10T00:00:00Z","timestamp":1602288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider the generic quadratic first integral (QFI) of the form I=Kab(t,q)q\u02d9aq\u02d9b+Ka(t,q)q\u02d9a+K(t,q) and require the condition dI\/dt=0. The latter results in a system of partial differential equations which involve the tensors Kab(t,q), Ka(t,q), K(t,q) and the dynamical quantities of the dynamical equations. These equations divide in two sets. The first set involves only geometric quantities of the configuration space and the second set contains the interaction of these quantities with the dynamical fields. A theorem is presented which provides a systematic solution of the system of equations in terms of the collineations of the kinetic metric in the configuration space. This solution being geometric and covariant, applies to higher dimensions and curved spaces. The results are applied to the simple but interesting case of two-dimensional (2d) autonomous conservative Newtonian potentials. It is found that there are two classes of 2d integrable potentials and that superintegrable potentials exist in both classes. We recover most main previous results, which have been obtained by various methods, in a single and systematic way.<\/jats:p>","DOI":"10.3390\/sym12101655","type":"journal-article","created":{"date-parts":[[2020,10,17]],"date-time":"2020-10-17T07:23:22Z","timestamp":1602919402000},"page":"1655","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Integrable and Superintegrable Potentials of 2d Autonomous Conservative Dynamical Systems"],"prefix":"10.3390","volume":"12","author":[{"given":"Antonios","family":"Mitsopoulos","sequence":"first","affiliation":[{"name":"Department of Astronomy-Astrophysics-Mechanics, Faculty of Physics, University of Athens, Panepistemiopolis, 157 83 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Tsamparlis","sequence":"additional","affiliation":[{"name":"Department of Astronomy-Astrophysics-Mechanics, Faculty of Physics, University of Athens, Panepistemiopolis, 157 83 Athens, Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9966-5517","authenticated-orcid":false,"given":"Andronikos","family":"Paliathanasis","sequence":"additional","affiliation":[{"name":"Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa"},{"name":"Instituto de Ciencias F\u00edsicas y Matem\u00e1ticas, Universidad Austral de Chile, Valdivia 5090000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1630009","DOI":"10.1142\/S0219887816300099","article-title":"Nonlinear ordinary differential equations: A discussion on symmetries and singularities","volume":"13","author":"Paliathanasis","year":"2016","journal-title":"Int. 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