Planar circular restricted 3-body problem

Question

Hi and sorry for my bad English, it's not my first language. I'm trying to find the equations of motion of the planar circular restricted 3-body problem. I did the gravitational force, but I have some problems with the centrifugal force. The vector equation is $F_{\text{cen}}=-m\vec{w} \times (\vec{w} \times \vec{r}))$, where $\vec{w}$ is the angular velocity and $\times$ is for the vector product. My problem is I'm trying to find the equations with $x$ and $y$, but with no $z$ coordinates, like in this paper : http://www.fundp.ac.be/pdf/publications/61801.pdf, which is in French. I use the same system of coordinates (p.6 (on the paper) and the b) one) and his equations only have $x$ and $y$ coordinates : $x''=2y' + x -Fg_1-Fg_2$ and $y''=-2x' + y -(Fg_1+Fg_2)$ When I do the cross product in the centrifugal force's equation, I have z coordinates for the y acceleration. I don't understand if I'm supposed to use the vector equation or not and if yes, then I don't understand what I'm supposed to do with the z coordinates. I'm only looking for advices. Thank you!