  This article aims to investigate the points of equilibrium and the associated
convergence basins in a seventh-order generalized H\'enon-Heiles potential.
Using the well-known Newton-Raphson iterator we numerically locate the position
of the points of equilibrium, while we also obtain their linear stability.
Furthermore, we demonstrate how the two variable parameters, entering the
generalized H\'enon-Heiles potential, affect the convergence dynamics of the
system as well as the fractal degree of the basin diagrams. The fractal degree
is derived by computing the (boundary) basin entropy as well as the uncertainty
dimension.
