The discussion of the periodic fixed points of Bäcklund transformations for the Korteweg–de Vries equation is completed. It will be shown that the systems of equations defined by the KdV periodic fixed points are e q u i v a l e n t to the periodic Kac–Van Moerbeke systems. As a consequence, for even order fixed points, the KdV systems are equivalent to the periodic Toda lattice. The periodic fixed points of the Bäcklund transformation for the Boussinesq equation are found to have a Hamiltonian structure. The integrals of these systems are found.