We investigate two topics regarding solar mass FGK-type stars, the lithium rotation connection (LRC) and the existence of the lithium desert. We determine the minimum critical rotation velocity ($v sin i$) related with the LRC separating slow from rapid stellar rotators, as being 5 km s$^{-1}$. This value also split different stellar properties. For the first time we explore the behaviour of the LRC for some stellar associations with ages between 45 Myr and 120 Myr. This allows us to study the LRC age dependence at the beginning of the general spin down stage for low mass stars, which starts at $sim$ 30-40 Myr. We find that each stellar group presents a characteristic minimum lithium (Li) depletion connected to a specific large rotation velocity and that this minimum changes with age. For instance, this minimum changes from $sim$ 50 km s$^{-1}$ to less than 20 km s$^{-1}$ in 200 Myr. Regarding the lithium desert, it was described as a limited region in the A(Li)-$T_{rm eff}$ map containing no stars. Using $T_{rm eff}$ from {em Gaia} DR2 we detect 30 stars inside and/or near the same box defined originally as the Li desert. Due to their intrinsic $T_{rm eff}$ errors some of these stars may be inside or outside the box, implying a large probability that the box contains several stars. Considering this last fact the lithium desert appears to be more a statistical distribution fluctuation than a real problem.
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