  We study the impact of classical short-range nonlinear interactions on
transport in lattices with no dispersion. The single particle band structure of
these lattices contains flat bands only, and cages non-interacting particles
into compact localized eigenstates. We demonstrate that there always exist
local unitary transformations that detangle such lattices into decoupled sites
in dimension one. Starting from a detangled representation, inverting the
detangling into entangling unitary transformations and extending to higher
lattice dimensions, we arrive at an All-Bands-Flat generator for single
particle states in any lattice dimension. The entangling unitary
transformations are parametrized by sets of angles. For a given member of the
set of all-bands-flat, additional short-range nonlinear interactions destroy
caging in general, and induce transport. However, fine-tuned subsets of the
unitary transformations allow to completely restore caging. We derive the
necessary and sufficient fine-tuning conditions for nonlinear caging, and
provide computational evidence of our conclusions for one-dimensional systems.
