Recently, a family of unconventional integrators for ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for quadratic vector fields. All these integrators seem to possess remarkable conservation properties. In particular, it has been proved that, when the underlying ODE is Hamiltonian, its polarization discretization possesses an integral of motion and an invariant volume form. In this note, we propose a new algebraic approach to derivation of the integrals of motion for polarization discretizations.

Comment: v2 re-formatted for the journal

• 37J70 - Completely integrable discrete dynamical systems
• 37M15 - Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
• 39A36 - Integrable difference and lattice equations; integrability tests
• 78-XX - Optics, electromagnetic theory
• 81-XX - Quantum theory