  Physical properties reflecting valley asymmetry of Landau levels in a biased
bilayer graphene under magnetic field are discussed. Within the $4-$band
continuum model with Hartree-corrected self-consistent gap and finite damping
factor we predict the appearance of anomalous steps in quantized Hall
conductivity due to the degeneracy lifting of Landau levels. Moreover, the
valley symmetry breaking effect appears as a non-semiclassical de Haas-van
Alphen effect where the reduction of the oscillation period to half cannot be
accounted for through quasi-classical quantization of the orbits in reciprocal
space, still valley degenerate.
