  We investigate Bloch-Zener oscillations and mean-field Bloch bands of a
Bose-Einstein condensate (BEC) in a Lieb optical lattice. We find that the
atomic interaction will break the point group symmetry of the system, leading
to the destruction of the Dirac cone structure, while the flat band is
preserved on the highly symmetric lines. Due to the nonlinear effect, a tubular
band structure with a flat band will appear in the system. Furthermore,
comparing with that the tight-binding (TB) model fails to describe the
interacting bosonic systems in the honeycomb lattice, we show that the TB model
is applicable to study the nonlinear energy band structures for the Lieb
lattice. In addition, we show that the loop structure can be determined by the
observation of the chaos of the state in the Bloch-Zener oscillations.
