We present results from an ongoing survey of searching Li-rich K giants among low mass giants along the Red Giant Branch (RGB). A sample of 2500 stars with accurate astrometry have been selected from Hipparcos catalogue covering both the RGB luminosity bump and the red clump regions on the HR diagram. Lithium abundances have been determined for half of the sample from low resolution spectra using line depth ratio method. Results confirm the rarity of Li-rich K giants, just under 1%, in the solar neighbourhood. This study increased the total number of known Li-rich K giants by a factor of two. The analysis of high resolution spectra of candidate Li-rich K giants showed that the K giant HD 77361 is highly enriched in lithium (log ϵ(Li) = 3.82) and at the same time has anomalously low carbon isotopic ratio (12C/13C = 4.3). The results put important constraints on the theoretical modelling of the stellar structure and the mixing process, particularly, of the K giants.