We have undertaken a detailed investigation of the lithium-for-manganese substitution effect of lithium manganese oxide, in the system LixMn3-xO4, for 0.95 < x < 1.05, that is for the nearly stoichiometric lithium content. The samples have been obtained by solid state reaction of Li2CO3 with the manganese oxide precursor, alpha-Mn2O3, at 750C and 800C. The lithium deficient samples were quenched in the solid CO2. High-resolution synchrotron X-ray measurements have been performed in the temperature range 20K-300K. The diffraction experiments were carried out at the DESY/HASYLAB powder diffractometer (beamline B2), equipped with a closed-cycle He cryostat. Very small changes in the lithium content influence clearly the low-temperature crystal structure of LixMn3-xO4 spinels and the nature of phase transitions. It was found that for x=0.95 the sample remains tetragonal in the whole 20K-300K temperature range. The stoichiometric LiMn2O4 transforms from cubic to orthorhombic at about 280K. For x=1.0125 the temperature of phase transition from cubic to orthorhombic decreases down to about 260K, whereas for x=1.025 the transformation goes from cubic directly to tetragonal, at the temperature 220K. No phase transition has been observed for the cubic sample with x=1.0375. Our results partly explain the divergences in recent reports on the low-temperature structure and phase transformations of lithium manganese oxides.