  We consider $tt^*$ equations appearing in the study of localized tachyon
condensations. They are described by various Toda system when we consider the
condensation by the lowest tachyon corresponding to the monomial $xy$. The
tachyon potential is calculated as a solution to these equations. The Toda
system appearing in the deformation of $\C^2/\Z_n$ by $xy$ is identical to that
of $D_n$ singularity deformed by $x$. For $\C^3/\Z_n$ with $xyz$ deformation,
we find only generic non-simple form, similar to the case appearing in
$\C/\Z_5\to \C/\Z_3$ and we discuss the difficulties in these cases.
