A conformal transformation \(f\) of a Riemannian manifold is essential, if the 1-parameter subgroup generated by \(f\) is not a group of isometries for any conformally equivalent metric. The main result of the paper deals with the behaviour of the flow generated by \(f\) near a fixed point. It turns out that the manifold under consideration is conformally flat in a neighborhood of the fixed point. The authors apply this result to the classification of Riemannian manifolds admitting twistor spinors with zeros and with non-vanishing associated vector field. These spaces are conformally flat, a result proved several years ago by K. Habermann under additional curvature assumptions.