  We study first BGG operators and their solutions on homogeneous conformal
geometries. We focus on conformal Killing tensors, conformal Killing--Yano
forms and twistor spinors in particular. We develop an invariant calculus that
allows us to find solutions explicitly using only algebraic computations. We
also discuss applications to holonomy reductions and conserved quantities of
conformal circles. We demonstrate our result on examples of homogeneous
conformal geometries coming mostly from general relativity.
