abstract: We discuss a description of overdetermined conformal operators as first terms in the BGG-sequence as developed by Cap-Slovak-Soucek and Calderbank-Diemer in 2001. This representation is used to rewrite the resulting systems of equations in closed form, namely as parallel sections of a linear connection H. 2008. Interesting situations come from the operators governing almost Einstein scales, twistor spinors and conformal Killing fieldsforms.