  We report results of systematic analysis of various modes in the flatband
lattice, based on the diamond-chain model with the on-site cubic nonlinearity,
and its double version with the linear on-site mixing between the two lattice
fields. In the single-chain system, a full analysis is presented, first, for
the single nonlinear cell, making it possible to find all stationary states,
viz., antisymmetric, symmetric, and asymmetric ones, including an exactly
investigated symmetry-breaking bifurcation of the subcritical type. In the
nonlinear infinite single-component chain, compact localized states (CLSs) are
found in an exact form too, as an extension of known compact eigenstates of the
linear diamond chain. Their stability is studied by means of analytical and
numerical methods, revealing a nontrivial stability boundary. In addition to
the CLSs, various species of extended states and exponentially localized
lattice solitons of symmetric and asymmetric types are studied too, by means of
numerical calculations and variational approximation. As a result, existence
and stability areas are identified for these modes. Finally, the linear version
of the double diamond chain is solved in an exact form, producing two split
flatbands in the system's spectrum.
