We consider the cubic system of differential equationsformulawhere pj(x; y); qj(x; y) 2 R[x; y] are homogeneous polynomials of degree j. The origin O(0; 0) is a singular point which is a center or a focus (fine focus) for (1). We study the problem of the center for cubic system (1) assuming that the system has irreducible invariant algebraic curves (algebraic solutions). The problem of the center was solved: for cubic system (1) with four and three invariant straight lines [1-3]; for cubic system (1) with two invariant straight lines and one irreducible invariant conic [3]; for cubic system (1) with two invariant straight lines and one irreducible invariant cubic [4]; for a nineparameter cubic system that can be reduced to a Li´enard type system [5]. In this talk we consider the following problems: (i) determine the subclass of cubic differential systems (1) which has a given number of invariant algebraic curves of degrees one and three; (ii) for this subclass find the conditions under which the origin is a center.