  R. Hirota and K. Kimura discovered integrable discretizations of the Euler
and the Lagrange tops, given by birational maps. Their method is a
specialization to the integrable context of a general discretization scheme
introduced by W. Kahan and applicable to any vector field with a quadratic
dependence on phase variables. We report several novel observations regarding
integrability of the Kahan-Hirota-Kimura discretization. For several of the
most complicated cases for which integrability is known (Clebsch system,
Kirchhoff system, and Lagrange top),
  - we give nice compact formulas for some of the more complicated integrals of
motion and for the density of the invariant measure, and
  - we establish the existence of higher order Wronskian Hirota-Kimura bases,
generating the full set of integrals of motion.
  While the first set of results admits nice algebraic proofs, the second one
relies on computer algebra.
