  One-dimensional all-bands-flat lattices are networks with all bands being
flat and highly degenerate. They can always be diagonalized by a finite
sequence of local unitary transformations parameterized by a set of angles
\(\theta_{i}\). In our previous work, Ref.~\onlinecite{lee2023critical}, we
demonstrated that quasiperiodic perturbations of the one-dimensional
all-bands-flat lattice with \(\theta_{i} = \pi/4\) give rise to a
critical-to-insulator transition and fractality edges separating critical from
localized states. In this study we consider the full range of angles
\(\theta_{i}\)s available for the all-bands-flat model and study the effect of
the quasiperiodic perturbation. For weak perturbation, we derive an effective
Hamiltonian and we identify the sets of \(\theta_{i}\)s for which the effective
model maps to extended or off-diagonal Harper models and hosts critical states.
For all the other values of the angles the spectrum is localized. Upon
increasing the perturbation strength, the extended Harper model evolves into
the system with energy dependent critical-to-insulator transitions, that we dub
\emph{fractality edges}. The case where the effective model maps onto the
off-diagonal Harper model features a critical-to-insulator transition at a
finite disorder strength.
