  As a unified theory of integer and fractional quantum Hall plateau
transitions, a nonperturbative theory of the two-parameter scaling
renormalization group function is presented. By imposing global symmetries
known as ``the law of corresponding states'', we seek a possible form of
renormalization group flows. Asking for consistency with the result from
weak-localization perturbation theory, such restriction is so intense that we
can analytically determine its concrete form. Accordingly, the critical
exponent $\nu$ and the irrelevant scaling index $y$ are obtained analytically
and turn out to be irrational. Their values ($\nu\approx 2.1$, $y\approx 0.3$)
agree favorably well with experiments and numerics.
