  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.
