  We present a detailed analysis of a general relativistic static spherical
symmetric distribution in which both the radial and tangential pressures follow
a master polytropic equation of state that generalizes the standard treatment
and avoids the appearance of singularities in the system. In particular, we
find the corresponding Lane-Emden equation and integrate it for a wide range of
values of the parameters involved. We explore the parameter space with the aim
to find the set of parameters leading to reasonable physical solutions. Also,
we considered the effect of spherically symmetric perturbations of the matter
variables in order to analyze the possible apparition of cracking within the
compact distribution.
