Let be a homogeneous polynomial in two variables such that on its graph , the origin of is an isolated umbilical point. In this paper, the behavior of the principal distributions around is studied in relation to the existence of other umbilical points than and the behavior of the gradient vector field of .
Naoya ANDO. "The behavior of the principal distributions around an isolated umbilical point." J. Math. Soc. Japan 53 (1) 237 - 260, January, 2001. https://doi.org/10.2969/jmsj/05310237