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We give a construction of completely integrable ($2n$)-dimensionalHamiltonian systems with symplectic brackets of the Lie-Poisson type (linear incoordinates) and with quadratic Hamilton functions. Applying to any such systemthe so called Kahan-Hirota-Kimura discretization scheme, we arrive at abirational ($2n$)-dimensional map. We show that this map is symplectic withrespect to a symplectic structure that is a perturbation of the originalsymplectic structure on $\mathbb R^{2n}$, and possesses $n$ independentintegrals of motion, which are perturbations of the original Hamilton functionsand 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 vectorfields, their Kahan-Hirota-Kimura discretizations also commute and share theinvariant symplectic structure and the $n$ integrals of motion. This paperextends our previous ones, arXiv:1606.08238 [nlin.SI] and arXiv:1607.07085[nlin.SI], where similar results were obtained for Hamiltonian systems with aconstant (canonical) symplectic structure and cubic Hamilton functions.

A construction of commuting systems of integrable symplectic birational maps. Lie-Poisson case