The differential equations describing the self-similar profiles of velocity moments in turbulent round jets are investigated by the Lie-symmetry method. From this, solutions are constructed, which depend on free functions. Truncated Taylor expansion of the free functions yields expressions for the profiles. If the Taylor expansion is truncated after the leading term, the classical Gaussian profile for the mean velocity is recovered, constituting the first rigorous derivation of this well-known empirical result. Comparison to DNS data shows good agreement of the derived expressions with the actual profiles.