  In 1962, astronomers Michel H\'enon and Carl Heiles studied orbits of stars
around centers of galaxies to determine the third integral of motion in
galactic dynamics. In order to do this, they reduced the system down to a
2-dimensional axisymmetric Hamiltonian system. Now this is known as the
H\'enon-Heiles (HH) System. Due to its apparent simplicity but extremely
complicated dynamical behavior, this system is currently a paradigm in
dynamical systems.
  In this paper, we perform a series expansion up to the seventh order of a
potential with axial and reflection symmetries. After some transformations,
this turns into the generalized H\'enon-Heiles (GHH) system where we separate
the fifth and seventh-order terms. We qualitatively analyze this system for
energies near the threshold between bounded and unbounded motion with
Poincar\'e sections and quantitatively analyze with Lyapunov Exponents. We find
that particles far from the critical energy demonstrate less chaos.
Additionally, the fifth-order term creates more regularity while the
seventh-order term does the opposite.
