Context: Lithium is a fragile element, which is easily destroyed in the stellar interior. The existence of lithium-rich giants still represents a challenge for stellar evolution models. Aims: We have collected a large database of high-resolution stellar spectra of 824 candidate thick-disk giants having 2,MASS photometry and proper motions measured by the Southern Proper-Motion Program (SPM). In order to investigate the nature of Li-rich giants, we searched this database for giants presenting a strong Li,I resonance line. Methods: We performed a chemical abundance analysis on the selected stars with the MOOG code along with proper ATLAS-9 model atmospheres. The iron content and atmospheric parameters were fixed by using the equivalent width of a sample of Fe lines. We also derive abundances for C, N, and O and measure or derive lower limits on the $^{12}$C/$^{13}$C isotopic ratios, which is a sensible diagnostic of the stars evolutionary status. Results: We detected five stars with a lithium abundance higher than 1.5, i.e. Li-rich according to the current definition. One of them (SPM-313132) has A(Li)$>$3.3 and, because of this, belongs to the group of the rare super Li-rich giants. Its kinematics makes it a likely thin-disk member and its atmospheric parameters are compatible with it being a 4,M$_odot$ star either on the red giant branch (RGB) or the early asymptotic giant branch. This object is the first super Li-rich giant detected at this phase. The other four are likely low-mass thick-disk stars evolved past the RGB luminosity bump, as determined from their metallicities and atmospheric parameters. The most evolved of them lies close to the RGB-tip. It has A(Li)$>$2.7 and a low $^{12}$C/$^{13}$C isotopic ratio, close to the cool bottom processing predictions.
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