We consider here real planar differential cubic systems of the form (x,y),(x,y)xPyQ==&& (1), where P, Q ϵ R[x, y] are real polynomials with cubic homogenities mentioned in the title. In [1] the classification of cubic systems with maximum number of invariant straight lines (i.e. 9) was done, whereas cubic systems with 8 invariant lines possessing either four or three distinct infinite singularities were studied in [2] and [3]. Our main goal is to classify the family of cubic systems with homogeneous cubic part (x3, 0), possessing 8 invariant straight lines including the line at infinity, all the lines considered with their multiplicities. This family possesses two distinct infinite singularities. As a result we proved a classification theorem (Main Theorem) of such systems depending on their geometric proprieties. Moreover, applying the algebraic method of invariants, developed by Sibirskii and his disciples, we constructed the necessary and sufficient conditions for the realization of each one of the possible 16 configurations of invariant straight lines.