Cubic differential systems with two affine real non-parallel invariant straight lines of maximal multiplicity

In this article we classify all differential real cubic systems possessing two affine real non-parallel invariant straight lines of maximal multiplicity. We show that the maximal multiplicity of each of these lines is at most three. The maximal sequences of multiplicities: $m(3,3;1)$, $m(3,2;2)$, $m(3,1;3)$, $m(2,2;3)$, $m_\infty(2,1;3)$, $m_\infty(1,1;3)$ are determined. The normal forms and the corresponding perturbations of the cubic systems which realize these cases are given.