A New Discrete Hénon-Heiles System

By considering the Darboux transformation for the third order Lax operator of the Sawada-Kotera hierarchy, we obtain a discrete third order linear equation as well as a discrete analogue of the Gambier 5 equation. As an application of this result, we consider the stationary reduction of the fifth order Sawada-Kotera equation, which (by a result of Fordy) is equivalent to a generalization of the integrable case (i) Hénon-Heiles system. Applying the Darboux transformation to the stationary flow, we find a Bäcklund transformation (BT) for this finite-dimensional Hamiltonian system, which is equivalent to an exact discretization of the generalized case (i) Hénon-Heiles system. The Lax pair for the system is 3 × 3, and the BT satisfies the spectrality property for the associated trigonal spectral curve. We also give an example of how the BT may be used as a numerical integrator for the original continuous Hénon-Heiles system.