In this work we present new features observed in the Saturated Absorption (SA) spectrum on the D 1 line of K. In an uncoated optical glass cell containing pure K atoms, excitation by circularly polarized pump beam produces an enhancement of the amplitude of the crossover resonances due to the hyperfine transitions starting from the ground state F g = 2. This effect appears to be much more relevant when K atoms are contained in a cell coated with an anti-relaxation film. Here, the crossover resonance in the F g = 2 set of transition is not observed experimentally with linearly polarized pump light, while in case of circular polarization its amplitude is significantly enlarged. The Light Induced Atomic Desorption (LIAD) effect strongly improves the intensities of the SA resonances observed in coated cell.