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The unusual nature of the single, rapidly rotating, lithium-rich K giant HDE 233517, which is currently undergoing significant mass loss, prompted a search for giants with similar properties. High-dispersion spectroscopic observations were obtained of HD 219025, a known lithium-rich infrared-excess giant, plus 39 stars from a list of G and K giants with excess far-infrared emission. The projected rotational velocities of the vast majority of infrared-excess giants appear to be similar to those of normal G and K giants. Six giants have lithium abundances at or above theoretical upper envelope values. The percentage of such stars in the sample of 39 infrared-excess giants is similar to that of normal giants. The three giants with the largest lithium abundances have previously been discovered. None of the sample of 39 giants have an Hα line similar to the broadened and very asymmetric line of HDE 233517. The star with optical properties most similar to HDE 233517 is HD 219025.