ABSTRACT
R. Hirota and K. Kimura discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is an application, in the integrable context, of a general discretization scheme introduced by W. Kahan for arbitrary vector fields 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), ∙ $ \bullet $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429470462/301849f2-53e7-4646-acba-6259bf733f01/content/inline-math1150.tif"/> we give nice compact formulas for some of the more complicated integrals of motion and for the density of the invariant measure, and ∙ $ \bullet $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429470462/301849f2-53e7-4646-acba-6259bf733f01/content/inline-math1151.tif"/> 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.
