Integrable symplectic maps via reduction of Bäcklund transformation

We discuss the stationary potential equations as illustrative examples to explain how to construct integrable symplectic maps via Bäcklund transformations. We first give a terse survey of Bäcklund transformations of the potential KdV equation and the potential fifth-order KdV equation. Then, using Jacobi–Ostrogradsky coordinates, we obtain canonical Hamiltonian forms of the stationary potential equations. Finally, we construct symplectic maps from the reduction of a Bäcklund transformation and verify that they are integrable.