  Fulde, Ferrell, Larkin, and Ovchinnikov (FFLO) predicted inhomogeneous
superconducting and superfluid ground states, spontaneously breaking
translation symmetries. In this Letter, we demonstrate that the transition from
the FFLO to the normal state as a function of temperature or increased Fermi
surface splitting is not a direct one. Instead the system has an additional
phase transition to a different state where pair-density-wave superconductivity
(or superfluidity) exists only on the boundaries of the system, while the bulk
of the system is normal. The surface pair-density-wave state is very robust and
exists for much larger fields and temperatures than the FFLO state.
