  We consider anisotropic long-range interacting spin systems in $d$
dimensions. The interaction between the spins decays with the distance as a
power law with different exponents in different directions: we consider an
exponent $d_{1}+\sigma_1$ in $d_1$ directions and another exponent
$d_{2}+\sigma_2$ in the remaining $d_2\equiv d-d_1$ ones. We introduce a low
energy effective action with non analytic power of the momenta. As a function
of the two exponents $\sigma_1$ and $\sigma_2$ we show the system to have three
different regimes, two where it is actually anisotropic and one where the
isotropy is finally restored. We determine the phase diagram and provide
estimates of the critical exponents as a function of the parameters of the
system, in particular considering the case of one of the two $\sigma$'s fixed
and the other varying. A discussion of the physical relevance of our results is
also presented.
