  Local configurational symmetry in lattice structures may give rise to
stationary, compact solutions, even in the absence of disorder and
nonlinearity. These compact solutions are related to the existence of flat
dispersion curves (bands). Nonlinearity can destabilize such compactons. One
common flat-band-generating system is the 1D cross-stitch model, in which
compactons were shown to exist for the photonic lattice with Kerr nonlinearity.
The compactons exist there already in the linear regime and are not generally
destructed by that nonlinearity. Smooth nonlinearity of this kind does not
allow doing a complete stability analysis for this chain. We consider a
discrete mechanical system with flat dispersion bands, in which the
nonlinearity exists due to impact constraints. There, one can use the concept
of the saltation matrix for the analytic construction of the monodromy matrix.
Besides, we consider a smooth nonlinear lattice with linearly connected
massless boxes, each containing two symmetric anharmonic oscillators. In this
model, the flat bands and discrete compactons also readily emerge. This system
also permits performing comprehensive stability analysis, at least in the
anti-continuum limit, due to the reduced number of degrees-of-freedom. In both
systems, there exist two types of localization. The first one is the complete
localization, and the second one is the more common exponential localization.
The latter type is associated with discrete breathers (DBs). Two principal
mechanisms for the loss of stability are revealed. The first one is the
possible internal instability of the symmetric and/or antisymmetric solution in
the individual unit cell of the chain. One can interpret this instability
pattern as internal resonance between the compacton and the DB. The other
mechanism is global instability related to resonance of the stationary solution
with the propagation frequencies.
