  We give a construction of completely integrable 4-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 4-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 $\mathbb R^4$, and
possess two independent integrals of motion, which are perturbations of the
original Hamilton functions. 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.
