Lazutkin coordinates of the maximal symmetric periodic orbits on the ellipse

In De Simoi J., Kaloshin V., Wei Q. "Dynamical spectral rigidity among $\mathbb{Z}_2$-symmetric strictly convex domains close to a circle" (Appendix B coauthored with H. Hezari) Ann. of Math. 186.1 (2017), pp. 277-314 deformational spectral rigidity of $\mathbb{Z}_2$ symmetric domains close to the circle has been shown. One of the steps of the proof was to express the maximal symmetric periodic orbits in the Lazutkin parametrization. Here using the action-angle variables we find the second order approximation of Lazutkin coordinates of the maximal symmetric periodic orbits on the ellipses.