Multi-dimensional Darboux type differential systems with quadratic nonlinearities

In the article the $n$-dimensional autonomous Darboux type differential systems with nonlinearities of the $2^{nd}$ degree are considered. With the aid of theorem on integrating factor the particular invariant $GL(n, \mathbb R)$-integrals are constructed as well as the first integrals of Darboux type for considered systems. These integrals represent the algebraic curves of the $1^{st}$ degree. The recurrence formula of particular invariant $GL(n,\mathbb R)$-integrals of the Darboux type differential system is found.