Conserved quantities of discretizations by polarization

Recently, a family of unconventional integrators for higher order 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 first order ODEs with quadratic vector fields. All these integrators possess remarkable conservation properties. In particular, for the first and the second order Hamiltonian ODEs, the discretization by polarization possesses an integral of motion and an invariant volume form. In this note, we extend our previously proposed algebraic approach to derivation of these integrals to discretizations of ODEs of an arbitrary order. For all orders $\ge 3$, these integrals are new.