We consider the real cubic system of differential equations formulaand the vector field  formulaassociated to this system. The critical point (0; 0) of system (1) is either a focus or a center. The problem of distinguishing between a center and a focus is called the problem of the center. The straight line  formula  is called invariant for (1) if there exists a polynomial K 2 C[x; y] such that the identity  formulaTheorem 1. The system (2) has at origin a center if and only if at least one of the following four sets of conditions is satisfied:  formulaIn each of the cases (i) ¡ (iv) the system (2) is Darboux integrable. Theorem 2. The cubic system (1) with two real parallel invariant straight lines of total multiplicity three has at origin a center if and only if the first five Lyapunov quantities vanish Lj = 0; j = 1; 5:Example.formulaFor this system the straight lines x¡1 = 0 and 2(6+ p 6)x¡5 = 0 are invariant. The line x ¡ 1 = 0 has multiplicity two. The first four Lyapunov quantities vanish: L1 = L2 = L3 = L4 = 0 and the five one no: L5 = 9(6 + p 6)=2 6= 0: Therefore, the origin is a focus of multiplicity five.