For the investigation of the formation mechanisms of vortices in the sphere wake for different fluid flow regimes (for $ 270 < Re < 3000 $) the direct numerical simulation and the Hussain's $ -\lambda_{2}$ vortex-eduction technique have been used. Two different types of the formation mechanisms of vortices for $ 270 < Re < 400 $ and for $ Re > 400 $ are demonstrated. For $ 270 < Re < 400 $ a periodical separation of the hairpin-shaped vortices is observed only from one edge of the cylindrical shear layer (surrounding the recirculating zone of the sphere wake) and the time-average lift/side and torque moment coefficients are not equal to zero. For $ Re > 400 $ the periodical separation of the hairpin-shaped vortices is observed from opposite edges of the shear layer alternatively and the time-average lift/side and torque moment coefficients are equal to zero. In addition the regular and irregular rotation of the cylindrical shear layer are observed for $ 360 < Re < 400 $ and for $ Re > 600 $ correspondingly. All stages of the extraction of the hairpin-shaped vortices from the recirculating zone of the wake are shown in detail.