On Integrability of Hirota–Kimura Type Discretizations

We give an overview of the integrability of the Hirota–Kimura discretizationmethod applied to algebraically completely integrable (a.c.i.) systems with quadratic vector fields. Along with the description of the basic mechanism of integrability (Hirota–Kimura bases), we provide the reader with a fairly complete list of the currently available results for concrete a.c.i. systems.