The McMillan map, originally proposed by Edwin McMillan in 1971, is a fundamental example of a nonlinear integrable mapping in dynamical systems. It has been widely studied in both mathematics and physics due to its elegant mathematical structure and its relevance to practical applications, notably in accelerator physics.

In this talk, we introduce and investigate a novel many-body generalization of the McMillan map that is integrable and symplectic. We demonstrate that this map is intimately connected to the Kaup-Newell hierarchy; in particular, it arises as a Bäcklund transformation for the restricted Kaup-Newell flows and shares their integrals of motion. By introducing root variables and employing Abel-Jacobi coordinates, we show that the discrete-time evolution corresponds to a shift on the associated Jacobivariety.