  In the standard model of charge density wave (CDW) transitions, the
displacement along a single phonon mode lowers the total electronic energy by
creating a gap at the Fermi level, making the CDW a metal--insulator
transition. Here, using scanning tunneling microscopy and spectroscopy and ab
initio calculations, we show that VS$_2$ realizes a CDW which stands out of
this standard model. There is a full CDW gap residing in the unoccupied states
of monolayer VS$_2$. At the Fermi level, the CDW induces a topological
metal-metal (Lifshitz) transition. Non-linear coupling of transverse and
longitudinal phonons is essential for the formation of the CDW and the full gap
above the Fermi level. Additionally, x-ray magnetic circular dichroism reveals
the absence of net magnetization in this phase, pointing to coexisting charge
and spin density waves in the ground state.
