Explore the third lecture in a minicourse on spectrum rigidity and joint integrability for Anosov systems on tori. Delve into the study of irreducible and non-conformal Anosov automorphisms A ∈ Sp(4, Z) and their symplectic diffeomorphisms. Examine the conditions under which the extremal symplectic bundle of a C^1-close symplectic diffeomorphism f is integrable, and how this relates to smooth conjugacy with A. Gain insights into the rigidity properties of Anosov diffeomorphisms on tori and the connections between geometric rigidity and dynamical spectral rigidity.