{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T15:43:38Z","timestamp":1760888618651,"version":"3.41.2"},"reference-count":12,"publisher":"AIP Publishing","issue":"12","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2011,12,1]]},"abstract":"<jats:p>In this paper, we consider the three-wave interaction system \\documentclass[12pt]{minimal}\\begin{document}$\\dot{x}=\\gamma x +\\delta y +z\\break -2 y^2, \\quad \\dot{y} =\\gamma y -\\delta x +2 x y, \\quad \\dot{z} =-2 z -2 z x,$\\end{document}x\u0307=\u03b3x+\u03b4y+z\u22122y2,y\u0307=\u03b3y\u2212\u03b4x+2xy,z\u0307=\u22122z\u22122zx, where \u03b3, \u03b4 are real parameters. Our main results are the complete characterization of all values of the parameters \u03b3, \u03b4 for which the three-wave interaction system admits either invariant algebraic surfaces or algebraic (polynomial or rational) first integrals.<\/jats:p>","DOI":"10.1063\/1.3672193","type":"journal-article","created":{"date-parts":[[2012,1,3]],"date-time":"2012-01-03T21:08:40Z","timestamp":1325624920000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Invariant algebraic surfaces for the reduced three-wave interaction system"],"prefix":"10.1063","volume":"52","author":[{"given":"Adam","family":"Mahdi","sequence":"first","affiliation":[{"name":"University of North Carolina at Charlotte 1 Mathematics Department, , Charlotte, North Carolina 28223, USA"},{"name":"AGH University of Science and Technology 2 , Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland"}]},{"given":"Claudia","family":"Valls","sequence":"additional","affiliation":[{"name":"Universidade Tecnica de Lisboa 3 Instituto Superior Tecnico, , Av. Rovisco Pais 1049-001, Lisboa, Portugal"}]}],"member":"317","published-online":{"date-parts":[[2011,12,30]]},"reference":[{"key":"2023062804014721300_c1","doi-asserted-by":"publisher","first-page":"L669","DOI":"10.1088\/0305-4470\/25\/11\/009","article-title":"Lie symmetries for the reduced three-wave interaction problem","volume":"25","year":"1992","journal-title":"J. Phys. 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