Detailed non-LTE and LTE analyses are made of two very strong Li K giants, HD 19745 and HD 39853, and of the strong Li K giant HD 787. The stronger star HD 19745 has recently been discovered to show very strong Li. These analyses are based on the observations done at ESO of the resonance and first secondary Li I lines at 6708 and 6104 A, respectively, and on the UV continua measured with the IUE. The non-LTE effects found for these stars are large. For the two very strong Li giants, the non-LTE abundances of Li are much larger than the typical interstellar Li abundance, which suggest that this kind of star may be a significant source of new Li in the Galaxy. We need to understand however, how these low-mass and low-rotation stars can produce such large quantities of Li.
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