Some Integrability Properties of the Kahan Method

The Kahan method for quadratic ordinary differential equations has recently received a renewed attention due to its integrability properties when applied to some classes of integrable differential equations (see for instance [3, 1] and references therein). We consider a family of (super) integrable quadratic ODEs generated by monomial vector fields, that we call Elementary Monomial Vector Fields (EMVFs). The EMVFs possess integrals which are not necessarily quadratic and the vector fields themselves need not be Hamiltonian. We show that, for certain choices of integer coefficients, the Kahan method is integrable when applied to quadratic EMVFs and we provide the modified integrals.