  The recent investigation of chains of Rydberg atoms has brought back the
problem of commensurate-incommensurate transitions into the focus of current
research. In 2D classical systems, or in 1D quantum systems, the commensurate
melting of a period-p phase with p larger than 4 is known to take place through
an intermediate floating phase where correlations between domain walls or
particles decay only as a power law, but when p is equal to 3 or 4, it has been
argued by Huse and Fisher that the transition could also be direct and
continuous in a non-conformal chiral universality class with a dynamical
exponent larger than 1. This is only possible however if the floating phase
terminates at a Lifshitz point before reaching the conformal point, a
possibility debated since then. Here we argue that this is a generic feature of
models where the number of particles is not conserved because the exponent of
the floating phase changes along the Pokrovsky-Talapov transition and can thus
reach the value at which the floating phase becomes unstable. Furthermore, we
show numerically that this scenario is realized in an effective model of the
period-3 phase of Rydberg chains in which hard-core bosons are created and
annihilated three by three: The Luttinger liquid parameter reaches the critical
value $p^2/8=9/8$ along the Pokrovsky-Talapov transition, leading to a Lifshitz
point that separates the floating phase from a chiral transition. Implications
beyond Rydberg atoms are briefly discussed.
