In this paper, a method of so-called ‘generalized normal sectors’ is developed for determining orbits in exceptional directions near high degenerate equilibria for which the well-known methods of normal sectors and Z-sectors are not applicable so that a more extensive class of degenerate equilibria can be discussed for qualitative properties. Our method proves to be effective in the study of high degenerate equilibria in a multi-molecular reaction system at infinity and in a ratio-dependent predator–prey system.