  We study the local geometry of irreducible parabolic geometries admitting
strongly essential flows; these are flows by local automorphisms with
higher-order fixed points. We prove several new rigidity results, and recover
some old ones for projective and conformal structures, which show that in many
cases the existence of a strongly essential flow implies local flatness of the
geometry on an open set having the fixed point in its closure. For almost
c-projective and almost quaternionic structures we can moreover show flatness
of the geometry on a neighborhood of the fixed point.
