Newton's method and a surface with Jacobian zero

Suppose a system of nonlinear real equations $P(x)=0$, where $P$ and $x$ are n-dimensional vectors, is solved by means of the continuous analog of Newton's method. We study the behavior of the method near the surface $S$ with Jacobian zero: $S=\{x|det P'(x)=0\}$. A computational strategy is suggested in the case where the method diverges.