  It is shown that, an entire class of off-diagonally disordered linear
lattices composed of two basic building blocks and described within a tight
binding model can be tailored to generate absolutely continuous energy bands.
It can be achieved if linear atomic clusters of an appropriate size are side
coupled to a suitable subset of sites in the backbone, and if the nearest
neighbor hopping integrals, in the backbone and in the side coupled cluster
bear a certain ratio. We work out the precise relationship between the number
of atoms in one of the building blocks in the backbone, and that in the side
attachment. In addition, we also evaluate the definite correlation between the
numerical values of the hopping integrals at different subsections of the
chain, that can convert an otherwise point spectrum (or, a singular continuous
one for deterministically disordered lattices) with exponentially (or power law
) localized eigenfunctions to an absolutely continuous spectrum comprising one
or more bands (subbands) populated by extended, totally transparent
eigenstates. The results, which are analytically exact, put forward a
non-trivial variation of the Anderson localization [P. W. Anderson, Phys. Rev.
109, 1492 (1958)], pointing towards its unusual sensitivity to the numerical
values of the system parameters and, go well beyond the other related models
such as the Random Dimer Model (RDM) [Dunlap et al., Phys. Rev. Lett. 65, 88
(1990)].
