  The dynamic equations for the fluxes through the SQUIDs that form a
two-dimensional metamamaterial on a Lieb lattice are derived, and then
linearized around zero flux to obtain the linear frequency spectrum according
to the standard procedure. That spectrum, due to the Lieb lattice geometry,
possesses a frequency band structure exhibiting two characteristic features;
two dispersive bands, which form a Dirac cone at the corners of the first
Brillouin zone, and a flat band crossing the Dirac points. It is demonstrated
numerically that localized states can be excited in the system when it is
initialized with single-site excitations; depending on the amplitude of those
initial states, the localization is either due to the flat-band or to nonlinear
effects. Flat-band localized states are formed in the nearly linear regime,
while localized excitations of the discrete breather type are formed in the
nonlinear regime. These two regimes are separated by an intermediate turbulent
regime for which no localization is observed. Notably, initial single-site
excitations of only edge SQUIDs of a unit cell may end-up in flat-band
localized states; no such states are formed for initial single-site excitations
of a corner SQUID of a unit cell. The degree of localization of the resulting
states is in any case quantified using well-established measures such as the
energetic participation ratio and the second moment.
