Lie algebras of operators and invariant $GL(2,\mathbb{R})$-integrals for Darboux type differential systems

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.