  We establish limit theorems for re-scaled occupation time fluctuations of a
sequence of branching particle systems in $\R^d$ with anisotropic space motion
and weakly degenerate splitting ability. In the case of large dimensions, our
limit processes lead to a new class of operator-scaling Gaussian random fields
with non-stationary increments. In the intermediate and critical dimensions,
the limit processes have spatial structures analogous to (but more complicated
than) those arising from the critical branching particle system without
degeneration considered by Bojdecki et al.{\it Stochastic Process. Appl. 2006}.
Due to the weakly degenerate branching ability, temporal structures of the
limit processes in all three cases are different from those obtained by
Bojdecki et al. .{\it Stochastic Process. Appl. 2006}.
