  We study vector fields generating a local flow by automorphisms of a
parabolic geometry with higher order fixed points. We develop general tools
extending the techniques of [1], [2], and [3]. We apply these tools to almost
Grassmannian, almost quaternionic, and contact parabolic geometries, including
CR structures, to obtain descriptions of the possible dynamics of such flows
near the fixed point and strong restrictions on the curvature. In some cases,
we can show vanishing of the curvature on a nonempty open set. Deriving
consequences for a specific geometry entails evaluating purely algebraic and
representation-theoretic criteria in the model homogeneous space. Published in
Indiana University Mathematics Journal.
