Feynman's proof of the commutativity of the Calogero integrals of motion

During his last year of life Feynman became interested in integrable models. In the course of his study of one dimensional particles with inverse square potentials (the Calogero model) he came up with a field theory-inspired proof of the commutativity of the integrals of motion of the system, which remained unpublished. I present an exact transcript of the proof together with a scan of the first page of Feynman's original manuscript and some historical background.