Pentagram Rigidity for Centrally Symmetric Octagons

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally symmetric octagons. This is the simplest case that goes beyond an analysis of elliptic curves. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail.
Comment: This paper is a fairly substantial revision of the previous version. I revised it for publication. The new version is generally cleaner and has fewer typos. Also, the endgame of the proof is simpler