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We consider the class of polynomial differential systems of the formẋ = λx − y + P n (x, y),ẏ = x + λy + Q n (x, y), where P n and Q n are homogeneous polynomials of degree n. For this class of differential systems we summarize the known results for the existence of limit cycles, and we provide new results for their non-existence and existence. 2010 Mathematics Subject Classification. 34C35; 34D30.doi:10.4153/cmb-2015-062-1 fatcat:msey6d5edjd4zm5pb2feevi63u