It is shown that in the old and celebrated problem of one-dimensional localization on a lattice with random on-site energies the joint probability distribution (JPD) of amplitudes and phases of random eigenfunction exhibits an anomaly at any rational filling factor. The principal anomaly at half-filling is described by the transfer-matrix equation with the two-dimensional (amplitude and phase) internal space and an explicit dependence on both co-ordinates. We show that the  zero mode variant of this equation is exactly solvable and find the unique solution for the JPD.