  Recently Hirota and Kimura presented a new discretization of the Euler top
with several remarkable properties. In particular this discretization shares
with the original continuous system the feature that it is an algebraically
completely integrable bi-Hamiltonian system in three dimensions. The
Hirota-Kimura discretization scheme turns out to be equivalent to an approach
to numerical integration of quadratic vector fields that was introduced by
Kahan, who applied it to the two-dimensional Lotka-Volterra system.
  The Euler top is naturally written in terms of the $\mathfrak{so}(3)$
Lie-Poisson algebra. Here we consider algebraically integrable systems that are
associated with pairs of Lie-Poisson algebras in three dimensions, as presented
by G\"umral and Nutku, and construct birational maps that discretize them
according to the scheme of Kahan and Hirota-Kimura. We show that the maps thus
obtained are also bi-Hamiltonian, with pairs of compatible Poisson brackets
that are one-parameter deformations of the original Lie-Poisson algebras, and
hence they are completely integrable. For comparison, we also present analogous
discretizations for three bi-Hamiltonian systems that have a transcendental
invariant, and finally we analyze all of the maps obtained from the viewpoint
of Halburd's Diophantine integrability criterion.
