  In anisotropic turbulence the correlation functions are decomposed in the
irreducible representations of the SO(3) symmetry group (with different
"angular momenta" $\ell$). For different values of $\ell$ the second order
correlation function is characterized by different scaling exponents
$\zeta_2(\ell)$. In this paper we compute these scaling exponents in a Direct
Interaction Approximation (DIA). By linearizing the DIA equations in small
anisotropy we set up a linear operator and find its zero-modes in the inertial
interval of scales. Thus the scaling exponents in each $\ell$-sector follow
from solvability condition, and are not determined by dimensional analysis. The
main result of our calculation is that the scaling exponents $\zeta_2(\ell)$
form a strictly increasing spectrum at least until $\ell=6$, guaranteeing that
the effects of anisotropy decay as power laws when the scale of observation
diminishes. The results of our calculations are compared to available
experiments and simulations.
