  The one-dimensional (1d) Anderson model (AM) has statistical anomalies at any
rational point $f=2a/\lambda_{E}$, where $a$ is the lattice constant and
$\lambda_{E}$ is the de Broglie wavelength. We develop a regular approach to
anomalous statistics of normalized eigenfunctions $\psi(r)$ at such
commensurability points. The approach is based on an exact integral
transfer-matrix equation for a generating function $\Phi_{r}(u, \phi)$ ($u$ and
$\phi$ have a meaning of the squared amplitude and phase of eigenfunctions, $r$
is the position of the observation point). The descender of the generating
function ${\cal P}_{r}(\phi)\equiv\Phi_{r}(u=0,\phi)$ is shown to be the
distribution function of phase which determines the Lyapunov exponent and the
local density of states.
  In the leading order in the small disorder we have derived a second-order
partial differential equation for the $r$-independent ("zero-mode") component
$\Phi(u, \phi)$ at the $E=0$ ($f=\frac{1}{2}$) anomaly. This equation is
nonseparable in variables $u$ and $\phi$. Yet, we show that due to a hidden
symmetry, it is integrable and we construct an exact solution for $\Phi(u,
\phi)$ explicitly in quadratures. Using this solution we have computed moments
$I_{m}=N<|\psi|^{2m}>$ ($m\geq 1$) for a chain of the length $N \rightarrow
\infty$ and found an essential difference between their $m$-behavior in the
center-of-band anomaly and for energies outside this anomaly. Outside the
anomaly the "extrinsic" localization length defined from the Lyapunov exponent
coincides with that defined from the inverse participation ratio ("intrinsic"
localization length). This is not the case at the $E=0$ anomaly where the
extrinsic localization length is smaller than the intrinsic one.
