  The H\'enon-Heiles potential was first proposed as a simplified version of
the gravitational potential experimented by a star in the presence of a
galactic center. Currently, this system is considered a paradigm in dynamical
systems because despite its simplicity exhibits a very complex dynamical
behavior. In the present paper, we perform a series expansion up to the
fifth-order of a potential with axial and reflection symmetries, which after
some transformations, leads to a generalized H\'enon-Heiles potential. Such new
system is analyzed qualitatively in both regimes of bounded and unbounded
motion via the Poincar\'e sections method and plotting the exit basins. On the
other hand, the quantitative analysis is performed through the Lyapunov
exponents and the basin entropy, respectively. We find that in both regimes the
chaoticity of the system decreases as long as the test particle energy gets far
from the critical energy. Additionally, we may conclude that despite the
inclusion of higher order terms in the series expansion, the new system shows
wider zones of regularity (islands) than the ones present in the H\'enon-Heiles
system.
