Beginning in the late 90th the present author established the notion that turbulent scaling laws are exact solutions of the multi-point correlation equations (MPCE) of invariant type based on Lie symmetries. Though solutions for arbitrary moments were generated only the mean velocities clearly followed the theory suggesting that key characteristics were missing. This gap was partially closed in 2010 with the finding of an infinite set of statistical symmetries and finally fully completed recently with the derivation of a further extended set of statistical symmetries valid for wall-bounded shear flows. Combining these symmetries the new solutions for the higher moments exhibit an excellent match for a broad variety of canonical wall bounded shear flows such as boundary layer, Poiseulle and Couette flow inlcuding those with rotation and transpiration. Other then originally suggested for the mean velocity, higher order moments do not follow the maximum symmetry principle. Instead certain statistical symmetries are only active if the mean velocity in the correspondent is non-zero and hence, these symmetries are of conditional nature.