Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariant.
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The planar quadratic systems having a Darboux invariant defined by invariant straight lines of total multiplicity two or by an invariant conic have been studied in  and , respectively. Here we shall present the normal forms of the planar quadratic systems having an invariant cubic. Moreover we classify the phase portraits in the Poincare disc of all planar quadratic polynomial differential systems with invariant cubic curve and having a Darboux invariant defined by it.