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00925njm 22002777a 4500 dc Giné, Jaume author Valls, Claudia author 2022-01-24T09:19:52Z 2022-01-24T09:19:52Z 2018 In this paper we study the existence of local analytic first integrals for complex polynomial differential systems of the form ẋ = x + Pn(x, y), ẏ = −y, where Pn(x,y) is a homogeneous polynomial of degree n, called the complex homogeneous Kukles systems of degree n. We characterize all the homogeneous Kukles systems of degree n that belong to the Sibirsky ideal. Finally, we provide necessary and sufficient conditions when n = 2,...,7 in order that the complex homogeneous Kukles system has a local analytic first integral computing the saddle constants and using Gröbner bases to find the decomposition of the algebraic variety into its irreducible components. The first author is partially supported by a MINECO/ FEDER grant number 2017-84383-P and an AGAUR (Generalitat de Catalunya) grant number 2017SGR 1276. The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013. http://hdl.handle.net/10459.1/72807 Integrability Complex center-focus problem Saddle constants Kukles systems Gröbner Basis Integrability Conditions for Complex Homogeneous Kukles Systems