  We investigate in detail the escape dynamics in an analytical gravitational
model which describes the motion of stars in a quasar galaxy with a disk and a
massive nucleus. We conduct a thorough numerical analysis distinguishing
between regular and chaotic orbits as well as between trapped and escaping
orbits, considering only unbounded motion for several energy levels. In order
to distinguish safely and with certainty between ordered and chaotic motion we
apply the Smaller ALingment Index (SALI) method. It is of particular interest
to locate the escape basins through the openings around the collinear
Lagrangian points $L_1$ and $L_2$ and relate them with the corresponding
spatial distribution of the escape times of the orbits. Our exploration takes
place both in the configuration $(x,y)$ and in the phase $(x,\dot{x})$ space in
order to elucidate the escape process as well as the overall orbital properties
of the galactic system. Our numerical analysis reveals the strong dependence of
the properties of the considered escape basins with the total orbital energy,
with a remarkable presence of fractal basin boundaries along all the escape
regimes. We hope our outcomes to be useful for a further understanding of the
escape mechanism in active galaxy models.
