We have undertaken a detailed near-IR spectroscopic analysis of eight notable white dwarfs, predominantly of southern declination. In each case the spectrum failed to reveal compelling evidence for the presence of a spatially unresolved, cool, late-type companion. Therefore, we have placed an approximate limit on the spectral-type of a putative companion to each degenerate. From these limits we conclude that if GD659, GD50, GD71 or WD2359-434 possesses an unresolved companion then most probably it is substellar in nature (M<0.072Msun). Furthermore, any spatially unresolved late-type companion to RE J0457-280, RE J0623-374, RE J0723-274 or RE J2214-491 most likely has M<0.082Msun. These results imply that if weak accretion from a nearby late-type companion is the cause of the unusual photospheric composition observed in a number of these degenerates then the companions are of very low mass, beyond the detection thresholds of this study. Furthermore, these results do not contradict a previously noted deficit of very-low-mass stellar and brown dwarf companions to main sequence F,G,K and early-M type primaries (a<1000AU).
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