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R. Hirota and K. Kimura discovered integrable discretizations of the Eulerand the Lagrange tops, given by birational maps. Their method is aspecialization to the integrable context of a general discretization schemeintroduced by W. Kahan and applicable to any vector field with a quadraticdependence on phase variables. According to a proposal by T. Ratiu,discretizations of the Hirota-Kimura type can be considered for numerousintegrable systems of classical mechanics. Due to a remarkable and not wellunderstood mechanism, such discretizations seem to inherit the integrabilityfor all algebraically completely integrable systems. We introduce anexperimental method for a rigorous study of integrability of suchdiscretizations. Application of this method to the Hirota-Kimura typediscretization of the Clebsch system leads to the discovery of fourfunctionally independent integrals of motion of this discrete time system,which turn out to be much more complicated than the integrals of the continuoustime system. Further, we prove that every orbit of the discrete time Clebschsystem lies in an intersection of four quadrics in the six-dimensional phasespace. Analogous results hold for the Hirota-Kimura type discretizations forall commuting flows of the Clebsch system, as well as for the $so(4)$ Eulertop.

On integrability of Hirota-Kimura type discretizations. Experimental study of the discrete Clebsch system