How can you construct the periodic orbits around libration points L1 and L2 in the circular restricted three-body problem and prove their existence?

Following up on a previous question about the classification of periodic orbits, how can they be constructed, especially the planar Lyapunov family, around libration points $L_1$ and $L_2$? And how can we prove their existence around those points?

Most papers I have found focus mainly on Halo orbits (see example) but I haven't found any that focus on the planar Lyapunov family.