The starting point for this paper lies in the results obtained by Tatsumi(2004)for isotropic turbulence with theself-preserving hypothesis.A careful consideration of the mathematical structure of the one-point velocity distributionfunction equation obtained by Tatsumi(2004)leads to an exact.analysis of all possible cases and to all admissiblesolutions of the problem.This paper revisits this interesting problem from a new point of view,and obtains a newcomplete set of solutions.Based on these exact solutions,some physically significant consequences of recent advancesin the theory of homogenous statistical solution of the Navier-Stokes equations are presented.The comparison withformer theory was also made.The origin of non-Gaussian character could be deduced from the above exact solutions