  The exceptional configuration of the minimal resolution $\hat{S}_G $ of a
Kleinian quotient surface $S_G (:= \CZ^2/G)$ is depicted by a $A$-$D$-$E$
Coxeter-Dynkin diagram. In this article, we show that branching indices of the
affine $A$-$D$-$E$ diagram is geometrically characterized by a certain special
function $F$ of $S_G$ as the multiplicities of its divisor components in
$\hat{S}_G$, a version parallel to the elliptic fibration near certain types of
simple singular fibers in Kodaira's elliptic surface theory. We further obtain
the uniqueness property of the function $F$ (modular local units) among all
local functions in $S_G$ near the singular point whose divisors in $\hat{S}_G $
display the affine $A$-$D$-$E$ diagram configuration.
