An ongoing SAO RAS program for the 6-m telescope to search for and study luminous blue variable (LBV) candidates in the M31 and M33 galaxies was started in 1997. This paper summarizes the results of our observations over the last 20 years. Over this time, we have a general lists of candidates and obtained photometry and spectroscopy for approximately one hundred stars from the list. A detailed study and classification of LBV candidates were carried out.We alsomonitored spectrally and photometrically the confirmed LBV stars in these galaxies. Based on these results, the spectral energy distributions (SEDs) of LBVs and LBV candidates were computed, and the interstellar extinction values, temperature, and luminosity were derived. We propose a new method of confirming the nature of LBV stars. It is based on a special property exhibited only by LBV stars—the constancy of the bolometric luminosity, with except of a rare phase of giant eruption, registered for two historical LBVs in our galaxy and about few dozen extragalactic LBVs. Using this method, we determine the fundamental parameters of these stars: temperature, radius, luminosity, and interstellar extinction. We discuss the nature of several individual LBV stars discovered by us in M33 and the classification of the LBV candidates in M31. Thus, for 20 years of observations we have found four LBV stars in M33; in the M31 galaxy with different methods, we have confirmed seven objects as a LBV class.
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