  This paper deals with the derivation and analysis of a seventh-order
generalization of the H\'enon-Heiles potential. The new potential has axial and
reflection symmetries, and finite escape energy with three channels of escape.
Based on SALI indicator and exits basins, the dynamic behavior of the
seventh-order system is investigated qualitatively in cases of bounded and
unbounded movement. Moreover, a quantitative analysis is carried out through
the percentage of chaotic orbits and the basin entropy, respectively. After
classifying large sets of initial conditions of orbits for several values of
the energy constant in both regimes, we observe that when the energy moves away
from the critical value, the chaoticity of the system decreases and the basin
structure becomes simpler with sharper and well defined bounds. Our results
suggest that when the seventh-order contributions of the potential are taken
into account, the system becomes less ergodic in comparison with the classical
version of the H\'enon-Heiles system.
