Can we understand the branching of reaction valleys for more than two degrees of freedom?

The model of the chemical reaction path is fundamental in Chemistry. We usually understand it as the pathway along a valley of the potential energy surface (PES). However, often a valley bifurcation occurs. This is controlled by a valley-ridge-inflection point (VRI). Up to now, 2-dimensional (2D) figures of a PES govern our understanding. But for more degrees of freedom, this might be misleading. Here, we explain the matter over a 3D configuration space; the PES is then a 3D hypersurface in an R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^4$$\end{document}. In this case a visualization is possible. We still can project curves on the PES down into the R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^3$$\end{document} of the configuration space. A method for the calculation of Newton trajectories (NT) is applied, because NTs bifurcate at VRI points. The example used is a simple mathematical test function. It ends with a threefold combination manifold of three 1D VRI lines. The intersection of the VRI lines forms a super-VRI point. The corresponding singular NT at the super-VRI point has eight branches.