The problem of the center for cubic differential systems with the line at infinity and an affine real invariant straight line of total algebraic multiplicity five

In this article, we study the real planar cubic differential systems with a non-degenerate monodromic critical point $M_0.$ In the cases when the algebraic multiplicity $m(Z)= 5$ or $m(l_1)+m(Z)\ge 5,$ where $Z=0$ is the line at infinity and $l_1=0$ is an affine real invariant straight line, we prove that the critical point $M_0$ is of the center type if and only if the first Lyapunov quantity vanishes. More over, if $m(Z)=5$ (respectively, $m(l_1)+m(Z)\ge 5,~ m(l_1)\ge j,~ j=2,3 $) then $M_0$ is a center if the cubic systems have a polynomial first integral (respectively, an integrating factor of the form $1/l_1^j$).