We consider the real cubic system of differential equations dx / dt = P(x, y), dy / dt = Q(x, y); GCD(P,Q) = 1, (1) max{deg(P), deg(Q)}= 3. A straight line a x + b y +g = 0, a,b ,g ÎC, (a,b ) ¹ (0,0) is invariant for (1) if there exists a polynomial K(x, y) such that the identity aP(x, y)+bQ(x, y) º (ax +by +g )K(x, y) holds. In this paper we show that if the cubic system admits an affine straight line of multiplicity six, then the line at infinity cannot have multiplicity greater than one. Theorem. Any cubic system having two invariant straight lines (including the line at infinity) of the multiplicity m(6,1) via affine transformation and time rescaling can be brought to the form , 1 3 , 0. 3 2 x& = x y& = + ax + x y a ¹ (2) For system (2), x = 0 is the invariant straight line of multiplicity m1 = 6 and the line at infinity has multiplicity m2 = 1 . The perturbed cubic system (3) has six distinct invariant straight lines = 0, 6 e + (3 - 3e ) - 3e - 3 3 x y x 2 0, 6 (3 3 ) 3 2 0, 6 (3 3 ) 6 2 0, 3 3 3 3 3 3 3 - e = ye + - e x + e - e = ye + - e x - e - e = 2 2 2 2 0 2 2 3 3 2 3 x - x e - xe - xe -e - ae = which converge to x = 0 when e ®0.