  We have developed an approach allowing us to resolve the problem of
non-conventional Anderson localization emerging in bilayered
periodic-on-average structures with alternating layers of right-handed and
left-handed materials. Recently, it was numerically discovered that in such
structures with weak fluctuations of refraction indices, the localization
length $L_{loc}$ can be enormously large for small wave frequencies $\omega$.
Within the fourth order of perturbation theory in disorder, $\sigma^2 \ll 1$,
we derive the expression for $L_{loc}$ valid for any $\omega$. In the limit
$\omega \rightarrow 0$ one gets a quite specific dependence, $L^{-1}_{loc}
\propto \sigma ^4 \omega^8$. Our approach allows one to establish the
conditions under which this effect can be observed.
