  BGG-equations are geometric overdetermined systems of PDEs on parabolic
geometries. Normal solutions of BGG-equations are particularly interesting and
we give a simple formula for the necessary and sufficient additional
integrability conditions on a solution. We then discuss a procedure for
coupling known solutions of BGG-equations to produce new ones. Employing a
suitable calculus for conformal spin structures this yields explicit coupling
formulas and conditions between almost Einstein scales, conformal Killing forms
and twistor spinors. Finally we discuss a class of generic twistor spinors that
provides an invariant decomposition of conformal Killing fields.
