  Spherically symmetric relativistic stars with the polytropic equation of
state (EoS), which possess the local pressure anisotropy, are considered within
the framework of general relativity. The generalized Lane-Emden equations are
derived for the arbitrary anisotropy parameter $\Delta=p_t-p_r$ ($p_t$ and
$p_r$ being the transverse and radial pressure, respectively). They are then
applied to some special ansatz for the anisotropy parameter in the form of the
differential relation between the anisotropy parameter $\Delta$ and the metric
function $\nu$. The analytical solutions of the obtained equations are found
for incompressible fluid stars and then used for getting their mass-radius
relation, gravitational and binding energy. Also, following the Chandrasekhar
variational approach, the dynamical stability of incompressible anisotropic
fluid stars with the polytropic EoS against radial oscillations is studied. It
is shown that the local pressure anisotropy with $p_t>p_r$ can make the
incompressible fluid stars unstable with respect to radial oscillations, in
contrast to incompressible isotropic fluid stars with the polytropic EoS which
are dynamically stable.
