  We revisit the question concerning stability of nonuniform superfluid states
of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type to thermal and quantum
fluctuations. Invoking the properties of the putative phase diagram of
two-component Fermi mixtures, on general grounds we argue, that for isotropic,
continuum systems the phase diagram hosting a long-range-ordered FFLO-type
phase envisaged by the mean-field theory cannot be stable to fluctuations at
any temperature $T>0$ in any dimensionality $d<4$. In contrast, in layered
unidirectional systems the lower critical dimension for the onset of FFLO-type
long-range order accompanied by a Lifshitz point at $T>0$ is $d=5/2$. In
consequence, its occurrence is excluded in $d=2$, but not in $d=3$. We propose
a relatively simple method, based on nonperturbative renormalization group to
compute the critical exponents of the thermal $m$-axial Lifshitz point
continuously varying $m$, spatial dimensionality $d$ and the number of order
parameter components $N$. We point out the possibility of a robust, fine-tuning
free occurrence of a quantum Lifshitz point in the phase diagram of imbalanced
Fermi mixtures.
