  This paper explored the physical acceptability conditions for anisotropic
matter configurations in General Relativity. The study considered a generalized
polytropic equation of state $P=\kappa {\rho}^{\gamma}+\alpha \rho -\beta$ for
a heuristic anisotropy. We integrated the corresponding Lane-Emden equation for
several hundred models and found the parameter-space portion ensuring the
configurations' acceptability. Polytrope based on the total energy density are
more stable than those with baryonic density, and small positive local
anisotropies produce acceptable models. We also found that polytropic
configurations where tangential pressures are greater than radial ones are also
more acceptable. Finally, convective disturbances do not generate cracking
instabilities. Several models emerging from our simulations could represent
candidate of astrophysical compact objects.
