We study a single component (i.e., single valley, spin-polarized) two-dimensional electron gas with $C_{3v}$ symmetry tuned to a Van-Hove (VH) singularity. Generically, there may be either three or six VH points at the Fermi level, related to each other by symmetry. Using a renormalization group analysis, we show that when the effective interactions between electrons at the VH points are positive, the system is stable. In contrast, if the effective interactions are negative, the system develops an instability toward either pair density wave (PDW) or charge density wave (CDW) orders, depending on the anisotropy of the dispersion at the VH points. The PDW may have either a single wave vector or multiple wave vectors. The PDW phase with three coexisting wave vectors can support fractional $\frac{h}{6e}$ vortices. The interplay between the geometry of the Fermi surface and the singularity of the density of states is the key that enables PDW formation.

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• Charge density waves
• Pair density wave
• Van Hove singularity
• Two-dimensional electron gas
• Renormalization group