Consider the generic cuartic differential system ( x_ = P (x; y) ; y_ = Q(x; y) ; (1) where P;Q 2 R[x; y], max fdeg P; degQg = 4, GCD(P;Q) = 1. According to [1], if a polynomial differential system has enough invariant straight lines considered with their multiplicities, then we can construct a Darboux first integral. Moreover, the number of the invariant straight lines with their multiplicities affects the existence of the limit cycles [3], the center problem [2] and other qualitative properties of a polynomial differential system [4]. If yP4?xQ4 6 0, where P4 and Q4 are homogeneous polynomials of degree 4, then the system (1) has at the infinity an invariant straight line of the multiplicity at least one. We know that that the multiplicity of this invariant straight line is at most 10, according to [5]. In this work, algebraically, we obtain two systems with these properties, which, via affine transformations and time rescaling, were brought to the form (2).