We give a construction of completely integrable four-dimensional Hamiltonian systems with cubic Hamilton functions. Applying to the corresponding pairs of commuting quadratic Hamiltonian vector fields the so called Kahan–Hirota–Kimura discretization scheme, we arrive at pairs of birational four-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on , and possess two independent integrals of motion, which are perturbations of the original Hamilton functions and which are in involution with respect to the perturbed symplectic structure. Thus, these maps are completely integrable in the Liouville–Arnold sense. Moreover, under a suitable normalization of the original pairs of vector fields, the pairs of maps commute and share the invariant symplectic structure and the two integrals of motion.
- Received July 4, 2016.
- Accepted January 18, 2017.
- © 2017 The Author(s)
Published by the Royal Society. All rights reserved.