  The charge instabilities of electron systems in the square lattice are
analyzed near the Van Hove singularity by means of a wilsonian renormalization
group approach. We show that the method preserves the spin rotational
invariance at all scales, allowing a rigorous determination of spin and charge
instabilities of the $t-t'$ Hubbard model. For $t'$ above $\approx 0.276 t$,
repulsive interactions fall into two different universality classes. One of
them has nonsingular response functions in the charge sector, while the other
is characterized by the splitting of the Van Hove singularity. At the level of
marginal perturbations, the Hubbard model turns out to be at the boundary
between the two universality classes, while extended models with
nearest-neighbor repulsive interactions belong to the latter class. In the case
of open systems allowed to exchange particles with a reservoir, we show the
existence of a range of fillings forbidden above and below the Van Hove
singularity. This has the property of attracting the Fermi level in the
mentioned range, as the system reaches its lowest energy when the Fermi energy
is at the singularity.
