  We propose a geometric construction of three-dimensional birational maps that
preserve two pencils of quadrics. The maps act as compositions of involutions,
which, in turn, act along the straight line generators of the quadrics of the
first pencil and are defined by the intersections with quadrics of the second
pencil. On each quadric of the first pencil, the maps act as two-dimensional
QRT maps.
  While these maps are of a pretty high degree in general, we find geometric
conditions which guarantee that the degree is reduced to 3. The resulting
degree 3 maps are illustrated by two known and two novel Kahan-type
discretizations of three-dimensional Nambu systems, including the Euler top and
the Zhukovski-Volterra gyrostat with two non-vanishing components of the
gyrostatic momentum.
