The finiteness of the set of branch points of a spherical mapping of a narrowing saddle surface

We consider an oriented, finitely connected narrowing saddle surface $F\in C^2$ in $R^3$ on which the set of points of zero Gaussian curvature consists only of isolated points. It is proved that a spherical mapping of this surface can only have a finite number of branch points and the structure of the boundary of its spherical image is studied.