Geodesic
Lie algebras of operators and invariant $GL(2,\mathbb{R})$-integrals for Darboux type differential systems
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2006), pp. 3-16 
O. V. Diaconescu; M. N. Popa. Lie algebras of operators and invariant $GL(2,\mathbb{R})$-integrals for Darboux type differential systems. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2006), pp. 3-16. http://geodesic.mathdoc.fr/item/BASM_2006_3_a0/
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     title = {Lie algebras of operators and invariant $GL(2,\mathbb{R})$-integrals for {Darboux} type differential systems},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {3--16},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2006_3_a0/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this article two-dimensional autonomous Darboux type differential systems with nonlinearities of the $i^{th} (i=\overline{2,7})$ degree with respect to the phase variables are considered. For every such system the admitted Lie algebra is constructed. With the aid of these algebras particular invariant $GL(2,\mathbb{R})$-integrals as well as first integrals of considered systems are constructed. These integrals represent the algebraic curves of the $(i-1)^{th}(i=\overline{2,7})$ degree. It is showed that the Darboux type systems with nonlinearities of the $2^{nd}$, the $4^{th}$ and the $6^{th}$ degree with respect to the phase variables do not have limit cycles.

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