The Toda lattice is a completely integrable mechanical Hamiltonian system, in which particles interact according to nearest neighbor interaction law. We study deformations of the Toda Hamiltonian, and show that within certain classes of deformations, the integrability of the Toda lattice is rigid. The proof relies on the fact that high energy orbits of the Toda flow approximate billiard orbits in polytopes. This is a joint work with Vadim Kalsohin and Illya Koval.