In the course of the MUCHFUSS project we have recently discovered four radial velocity (RV) variable, hot (Teff $approx$ 80,000 - 110,000 K) post-asymptotic giant branch (AGB) stars. Among them, we found the first known RV variable O(He) star, the only second known RV variable PG 1159 close binary candidate, as well as the first two naked (i.e., without planetary nebula (PN)) H-rich post-AGB stars of spectral type O(H) that show significant RV variations. We present a non-LTE spectral analysis of these stars along with one further O(H)-type star whose RV variations were found to be not significant. We also report the discovery of an far-infrared excess in the case of the PG 1159 star. None of the stars in our sample displays nebular emission lines, which can be explained well in terms of a very late thermal pulse evolution in the case of the PG 1159 star. The missing PNe around the O(H)-type stars seem strange, since we find that several central stars of PNe have much longer post-AGB times. Besides the non-ejection of a PN, the occurrence of a late thermal pulse, or the re-accretion of the PN in the previous post-AGB evolution offer possible explanations for those stars not harbouring a PN (anymore). In case of the O(He) star J0757 we speculate that it might have been previously part of a compact He transferring binary system. In this scenario, the mass transfer must have stopped after a certain time, leaving behind a low mass close companion that could be responsible for the extreme RV shift of 107.0 $pm$ 22.0 km/s measured within only 31 min.
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