  We numerically investigate the solutions to the effective equations of the
Bianchi II model within the "improved" Loop Quantum Cosmology (LQC) dynamics.
The matter source is a massless scalar field. We perform a systematic study of
the space of solutions, and focus on the behavior of several geometrical
observables. We show that the big-bang singularity is replaced by a bounce and
the point-like singularities do not saturate the energy density bound. There
are up to three directional bounces in the scale factors, one global bounce in
the expansion, the shear presents up to four local maxima and can be zero at
the bounce. This allows for solutions with density larger than the maximal
density for the isotropic and Bianchi I cases. The asymptotic behavior is shown
to behave like that of a Bianchi I model, and the effective solutions connect
anisotropic solutions even when the shear is zero at the bounce. All known
facts of Bianchi I are reproduced. In the "vacuum limit", solutions are such
that almost all the dynamics is due to the anisotropies. Since Bianchi II plays
an important role in the Bianchi IX model and the Belinskii, Khalatnikov,
Lifshitz (BKL) conjecture, our results can provide an intuitive understanding
of the behavior in the vicinity of general space-like singularities, when
loop-geometric corrections are present.
