We extend results on time-rescaled occupation time fluctuation limits of the (d, α, β)-branching particle system (0 < α ≤ 2, 0 < β ≤ 1) with Poisson initial condition. The earlier results in the homogeneous case (i.e., with Lebesgue initial intensity measure) were obtained for dimensions d > α/β only, since the particle system becomes locally extinct if d ≤ α/β. In this paper we show that by introducing high density of the initial Poisson configuration, limits are obtained for all dimensions, and they coincide with the previous ones if d > α/β.We also give high-density limits for the systems with finite intensity measures (without high density no limits exist in this case due to extinction); the results are different and harder to obtain due to the non-invariance of the measure for the particle motion. In both cases, i.e., Lebesgue and finite intensity measures, for low dimensions [d < α(1 + β)/β and d < α(2 + β)/(1 + β), respectively] the limits are determined by non-Lévy self-similar stable processes. For the corresponding high dimensions the limits are qualitatively different: S (Rd )-valued Lévy processes in the Lebesgue case, stable processes constant in time on (0,∞) in the finite measure case. For high dimensions, the laws of all limit processes are expressed in terms of Riesz potentials. If β = 1, the limits are Gaussian. Limits are also given for particle systems without branching, which yields