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Multi-dimensional Darboux type differential systems with quadratic nonlinearities. (English) Zbl 1136.34035

Algebraic and invariant approaches to differential equations provide criteria for reductions to simpler forms. In this essential work, the autonomous \(n\)-dimensional \((n>1)\) system of first-order ODEs with quadratic nonlinearities are studied under the action of the group of centre-affine transformations. If the first-order system under investigation admits an (\(n-1\))-dimensional abelian Lie algebra of operators, then an integrating factor gives rise to an integral of the system. As a consequence, for such systems, the authors construct \(\mathrm{GL}(n, \mathbb R)\)-integrals and first integrals of Darboux type.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C14 Symmetries, invariants of ordinary differential equations
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