  Starting from the observation that a flying saucer is a nonholonomic
mechanical system whose 5-dimensional configuration space is a contact
manifold, we show how to enrich this space with a number of geometric
structures by imposing further nonlinear restrictions on the saucer's velocity.
These restrictions define certain `manoeuvres' of the saucer, which we call
`attacking,' `landing,' or `G2 mode' manoeuvres, and which equip its
configuration space with three kinds of flat parabolic geometry in five
dimensions. The attacking manoeuvre corresponds to the flat Legendrean contact
structure, the landing manoeuvre corresponds to the flat hypersurface type CR
structure with Levi form of signature (1,1), and the most complicated G2
manoeuvre corresponds to the contact Engel structure with split real form of
the exceptional Lie group G2 as its symmetries. A celebrated double fibration
relating the two nonequivalent flat 5-dimensional parabolic G2 geometries is
used to construct a `G2 joystick,' consisting of two balls of radii in ratio
1:3 that transforms the difficult G2 manoeuvre into the pilot's action of
rolling one of joystick's balls on the other without slipping nor twisting.
