Stable umbilical points of a W-congruence

The class of $W$-congruences is a central object of Projective Differential Geometry. Nevertheless, their singularities has not been extensively studied. In this paper we prove a characterization of $W$-congruences that allow us to study their umbilical points. We describe examples of isolated stable umbilical points of type $A_m$, $m\in \{1,2,3,4\}$, and discuss some partial results concerning the generalization of these results to any $m\in \mathbb{N}\cup \{\mathrm{\infty }\}$.