We report the existence of structurally stable pulse-like solutions in the vicinity of an inverted Hopf bifurcation. These localized structures correspond to droplets in first order phase transitions, where they are known to be unstable. We show that the stabilisation mechanism is a non variational effect, i.e. is due to the non existence of a "free-energy" to minimize in the instability problem we consider. We propose this mechanism as an explanation for the existence of localized waves in shear flows or in convection experiments in binary fluid mixtures