  Kahan discretization is applicable to any quadratic vector field and produces
a birational map which approximates the shift along the phase flow. For a
planar quadratic Hamiltonian vector field, this map is known to be integrable
and to preserve a pencil of cubic curves. Generically, the nine base points of
this pencil include three points at infinity (corresponding to the asymptotic
directions of cubic curves) and six finite points lying on a conic. We show
that the Kahan discretization map can be represented in six different ways as a
composition of two Manin involutions, corresponding to an infinite base point
and to a finite base point. As a consequence, the finite base points can be
ordered so that the resulting hexagon has three pairs of parallel sides which
pass through the three base points at infinity. Moreover, this geometric
condition on the base points turns out to be characteristic: if it is
satisfied, then the cubic curves of the corresponding pencil are invariant
under the Kahan discretization of a planar quadratic Hamiltonian vector field.
