We have used the Australia Telescope National Facility Mopra 22-m antenna to search for 37.7-GHz (7(-2) - 8(-1}E) methanol masers towards a sample of thirty six class II methanol masers. The target sources are the most luminous class II methanol masers not previously searched for this transition, with isotropic peak 12.2-GHz maser luminosity greater than 250 Jy/kpc^2 and isotropic peak 6.7-GHz maser luminosity greater than 800 Jy/kpc^2. Seven new 37.7-GHz methanol masers were detected as a result of the search. The detection rate for 37.7-GHz methanol masers towards a complete sample of all such class II methanol maser sites south of declination -20 deg is at least 30 percent. The relatively high detection rate for this rare methanol transition is in line with previous predictions that the 37.7-GHz transition is associated with a late stage of the class II methanol maser phase of high-mass star formation. We find that there is a modest correlation between the ratio of the 6.7- and 37.7-GHz maser peak intensity and the 6.7- and 12.2-GHz maser peak intensity (correlation coefficient 0.63 in a log-log plot). We detected one new 38.3-GHz (6(2) - 5(3)A-) methanol maser towards G335.789+0.174. This is only the fourth source for which maser emission has been detected in this transition and it is the only one for which emission is not also observed in the 38.5-GHz 6(2) - 5(3)A+ transition.
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