The periodic fixed points of baecklund transformations

The periodic fixed points of the Baecklund transformations are finite dimensional invariant manifolds for the flow of the system. The dynamics occur as commuting hamiltonian flows on this finite dimensional manifold. We examine the flow of the KdV periodic fixed points in the neighborhood of steady states and reductions. These are analogous to a flow in the neighborhood of a sequence of heteroclinic points. The periodic fixed points of Backlund transformations also define a natural factorization of the two-dimensional Toda lattice as commuting hamiltonian flows. These have an interesting connection with caustic surfaces and the Laplace-Darboux transformation