As the theory of chaotic scattering in high-dimensional systems is poorly developed, it is very difficult to determine initial conditions for which interesting scattering events, such as long delay times, occur. We propose to use symmetry breaking as a way to gain the insight necessary to determine low-dimensional subspaces of initial conditions in which we can find such events easily. We study numerically the planar scattering off a disk moving on an elliptic Kepler orbit, as a simplified model of the elliptic restricted three-body problem. When the motion of the disk is circular, the system has an integral of motion, the Jacobi integral, which is no longer conserved for nonvanishing eccentricity. In the latter case, the system has an effective five-dimensional phase space and is therefore not amenable for study with the usual methods. Using the symmetric problem as a starting point we define an appropriate two-dimensional subspace of initial conditions by fixing some coordinates. This subspace proves to be useful to define scattering experiments where the rich and nontrivial dynamics of the problem is illustrated. We consider in particular trajectories which take very long before escaping or are trapped by consecutive collisions with the disk.