We consider the cubic system of differential equations x˙ = y + p2(x, y) + p3(x, y), y˙ = −x + q2(x, y) + q3(x, y), (1)where pj(x, y), qj (x, y) ∈ R[x, y] are homogeneous polinomials of degree j. The origin O(0, 0) is a singular point for (1) with purely imaginary eigenvalues, i.e. a focus or a center. The problem of distinguishing between a center and a focus (the problem of the center) is open for general cubic systems. In [1] the problem of the center was solved for cubic system (1) with: four invariant straight lines; three invariant straight lines; two invariant straight lines and one irreducible invariant conic. The center conditions for a cubic differential system (1) with two invariant straight lines and one irreducible invariant cubic curve Φ ≡ a30x3 + a21x2y + a12xy2 + a03y3 + x2 + y2 = 0 were found in [2] and for cubic system (1) having an integrating factor μ−1 = Φh were found in [3], where a30,a21,a12,a03 and h are real parameters. In this talk we give the conditions under which the cubic system (1) has an integrating factor of the form μ−1 = Ψh, (1) where Ψ ≡ a20x2+a11xy+a02y2+a10x+a01y+1 = 0 is an irreducible invariant conic and a20,a11,a02,a10,a01 and h are real parameters. According to [2] the cubic differential systems (1) which have integrating factors of the form (1) have a center at the singular point O(0, 0).