{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T09:12:43Z","timestamp":1775121163301,"version":"3.50.1"},"reference-count":13,"publisher":"AIP Publishing","issue":"3","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2011,3,1]]},"abstract":"<jats:p>We give a complete characterization of the Darbouxian first integrals of a generalized Raychaudhuri equation which appears in modern string cosmology and which has the form \\documentclass[12pt]{minimal}\\begin{document}$\\dot{x} =-\\frac{1}{2} x^2 -\\alpha x -2(y^2 +z^2 -w^2)-2 \\beta , \\dot{y} =-(\\alpha +x) y -\\gamma , \\dot{z} =-(\\alpha +x) z -\\delta , \\dot{w} =-(\\alpha +x) w$\\end{document}x\u0307=\u221212x2\u2212\u03b1x\u22122(y2+z2\u2212w2)\u22122\u03b2,y\u0307=\u2212(\u03b1+x)y\u2212\u03b3,z\u0307=\u2212(\u03b1+x)z\u2212\u03b4,w\u0307=\u2212(\u03b1+x)w, where \u03b1, \u03b2, \u03b3, \u03b4 are real parameters. Our approach uses the Darboux theory of integrability.<\/jats:p>","DOI":"10.1063\/1.3559065","type":"journal-article","created":{"date-parts":[[2011,3,4]],"date-time":"2011-03-04T16:30:38Z","timestamp":1299256238000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["Darbouxian integrals for generalized Raychaudhuri equations"],"prefix":"10.1063","volume":"52","author":[{"given":"Claudia","family":"Valls","sequence":"first","affiliation":[{"name":"Instituto Superior T\u00e9cnico Departamento de Matem\u00e1tica, , Av. Rovisco Pais 1049-001, Lisbon, Portugal"}]}],"member":"317","published-online":{"date-parts":[[2011,3,2]]},"reference":[{"key":"2023062804060385600_c1","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1017\/S0308210500030213","article-title":"Invariant algebraic curves and conditions for a center","volume":"124","year":"1994","journal-title":"Proc.-R. Soc. Edinburg, Sect. 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