  We give a construction of completely integrable ($2n$)-dimensional
Hamiltonian systems with symplectic brackets of the Lie-Poisson type (linear in
coordinates) and with quadratic Hamilton functions. Applying to any such system
the so called Kahan-Hirota-Kimura discretization scheme, we arrive at a
birational ($2n$)-dimensional map. We show that this map is symplectic with
respect to a symplectic structure that is a perturbation of the original
symplectic structure on $\mathbb R^{2n}$, and possesses $n$ independent
integrals of motion, which are perturbations of the original Hamilton functions
and are in involution with respect to the invariant symplectic structure. Thus,
this map is completely integrable in the Liouville-Arnold sense. Moreover,
under a suitable normalization of the original $n$-tuples of commuting vector
fields, their Kahan-Hirota-Kimura discretizations also commute and share the
invariant symplectic structure and the $n$ integrals of motion. This paper
extends our previous ones, arXiv:1606.08238 [nlin.SI] and arXiv:1607.07085
[nlin.SI], where similar results were obtained for Hamiltonian systems with a
constant (canonical) symplectic structure and cubic Hamilton functions.
