{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T20:57:26Z","timestamp":1770670646917,"version":"3.49.0"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T00:00:00Z","timestamp":1735257600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005992","name":"Ministry of Education and Science of the Republic of Bulgaria","doi-asserted-by":"publisher","award":["D01-325\/01.12.2023"],"award-info":[{"award-number":["D01-325\/01.12.2023"]}],"id":[{"id":"10.13039\/501100005992","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf\u2013Langford-type systems. First, by introducing cylindrical coordinates in its phase space, we show that the regarded system can be reduced to a two-dimensional Li\u00e9nard system, which corresponds to a second-order Li\u00e9nard equation. Then, we present (in explicit form) polynomial first and second integrals of Li\u00e9nard systems of the considered type identifying those values of their parameters for which these integrals exist. It is also proved that a generic Li\u00e9nard equation is factorizable if and only if the corresponding Li\u00e9nard system admits a second integral of a special form. It is established that each Li\u00e9nard system corresponding to a Hopf\u2013Langford system of the considered type admits such a second integral, and hence, the respective Li\u00e9nard equation is factorizable.<\/jats:p>","DOI":"10.3390\/axioms14010008","type":"journal-article","created":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T09:23:36Z","timestamp":1735291416000},"page":"8","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["First and Second Integrals of Hopf\u2013Langford-Type Systems"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0831-9396","authenticated-orcid":false,"given":"Vassil M.","family":"Vassilev","sequence":"first","affiliation":[{"name":"Institute of Mechanics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev St., Block 4, 1113 Sofia, Bulgaria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6147-1354","authenticated-orcid":false,"given":"Svetoslav G.","family":"Nikolov","sequence":"additional","affiliation":[{"name":"Institute of Mechanics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev St., Block 4, 1113 Sofia, Bulgaria"},{"name":"Department of Mechanics, University of Transport, 158, G. Milev St., 1574 Sofia, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1002\/cpa.3160010401","article-title":"A mathematical example displaying features of turbulence","volume":"1","author":"Hopf","year":"1948","journal-title":"Commun. Pure Appl. 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