  In this paper, we study a family of two-component anisotropic polytropes
which model a wide range of spherically symmetric astrophysical systems such as
early-type baryonic galaxies. This family is found to contain a large class of
models such as the hypervirial family (which satisfy the virial theorem
locally), the Plummer and Hernquist models and Navarro-Frenk-White (NFW)-like
models. The potential--density pair for these models are derived, as well as
their velocity dispersions and anisotropy parameters. The projected quantities
are computed and found to reduce to analytical expressions in some cases. The
first section of this paper presents an extension of the two-term anisotropic
polytropes to encompass a very wide range of potential-density pairs. In the
next section, we present the general relativistic extension of the
potential-density pair, and calculate the stress-energy tensor, the
relativistic anisotropy parameter, the velocity of circular orbits and the
angular momentum. Remarkably, for the case of the hypervirial family, the
relativistic pressure in the Newtonian limit and the relativistic anisotropy
parameter are found to coincide with the corresponding Newtonian expressions.
The weak, dominant and strong energy conditions are found to be satisfied only
for a certain range of the free parameters. We show that the relativistic
hypervirial family also has a finite total mass like its Newtonian counterpart.
In the first appendix, a relativistic extension of a different hypervirial
family of models is studied, and the relativistic anisotropy parameter is found
to coincide with the Newtonian one. Finally, we present a family of models
obtained from our distribution function that are similar to the
Ossipkov-Merritt models; by computing their anisotropy parameters, we show that
they model systems with isotropic cores and radially anisotropic exteriors.
