  We numerically explore the Newton-Raphson basins of convergence, related to
the libration points (which act as attractors of the convergence process), in
the generalized H\'{e}non-Heiles system (GHH). The evolution of the position as
well as of the linear stability of the equilibrium points is determined, as a
function of the value of the perturbation parameter. The attracting regions, on
the configuration $(x,y)$ plane, are revealed by using the multivariate version
of the classical Newton-Raphson iterative algorithm. We perform a systematic
investigation in an attempt to understand how the perturbation parameter
affects the geometry as well as of the basin entropy of the attracting domains.
The convergence regions are also related with the required number of
iterations.
