Normal forms and global phase portraits of quadratic and cubic integrable vector fields having two nonconcentric circles as invariant algebraic curves

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ABSTRACTIn this paper, we give the normal form of all planar polynomial vector fields of degree d ≤ 3 having two nonconcentric circles C1 and C2 as invariant algebraic curves and the function , with α and β real values, as first integral. Moreover, we classify all global phase portraits on the Poincaré disc of a subclass of these vector fields.