Integrability properties of Kahan's method (Q2922365)

In the paper under review, several novel examples of integrable quadratic vector fields for which Kahan's discretization method [\textit{W. Kahan} and \textit{R.-C. Li}, J. Comput. Phys. 134, No. 2, 316--331, Art. No. CP975710 (1997; Zbl 0883.65061)] preserves integrability are presented. Some examples include generalized Suslov and Ishii [\textit{M. Ishii}, Prog. Theor. Phys. 84, No. 3, 386--391 (1990; Zbl 1100.34535)] systems, Nambu systems, Riccati systems, and the first Painlevé equation. The paper also discusses how Manin transformations arise in Kahan discretizations of certain vector fields.NEWLINENEWLINEThe paper is organized as follows. Section 2 is on the application of Kahan's method to autonomous quadratic ODEs. Subsection 2.1 treats the discretization of quadratic Hamiltonian ODEs in \(\mathbb{R}^2\), and its connection to Manin transformations. Subsection 2.2 discusses the discretization of quadratic ODEs in \(\mathbb{R}^3\) possessing two quadratic integrals. Subsection 2.3 is about the discretization of generalized Suslov systems in \(\mathbb{R}^n\), including the integrable cases \(n = 2\) and \(n = 3\). Section 3 is on the application of Kahan's method to nonautonomous quadratic ODEs. Subsection 3.1 treats the discretization of certain Riccati equations. Last but not least, Section 3.2 covers the (non-autonomous) first Painlevé equation, whose Kahan discretization is also shown to be integrable. In this striking example the integrals are not rational functions and the mapping is not integrable in terms of elliptic or hyperelliptic functions.