  First BGG operators are a large class of overdetermined linear differential
operators intrinsically associated to a parabolic geometry on a manifold. The
corresponding equations include those controlling infinitesimal automorphisms,
higher symmetries, and many other widely studied PDE of geometric origin. The
machinery of BGG sequences also singles out a subclass of solutions called
normal solutions. These correspond to parallel tractor fields and hence to
(certain) holonomy reductions of the canonical normal Cartan connection. Using
the normal Cartan connection, we define a special class of local frames for any
natural vector bundle associated to a parabolic geometry. We then prove that
the coefficient functions of any normal solution of a first BGG operator with
respect to such a frame are polynomials in the normal coordinates of the
parabolic geometry. A bound on the degree of these polynomials in terms of
representation theory data is derived. For geometries locally isomorphic to the
homogeneous model of the geometry we explicitly compute the local frames
mentioned above. Together with the fact that on such structures all solutions
are normal, we obtain a complete description of all first BGG solutions in this
case. Finally, we prove that in the general case the polynominal system coming
from a normal solution is the pull-back of a polynomial system that solves the
corresponding problem on the homogeneous model. Thus we can derive a complete
list of potential normal solutions on curved geometries. Morover, questions
concerning the zero locus of solutions, as well as related finer geometric and
smooth data, are reduced to a study of polynomial systems and real algebraic
sets.
