  Inspired by a recent work of Dubrovin [7], for each simple Lie algebra
$\mathfrak{g}$, we introduce an infinite family of pairwise commuting ODEs and
define their $\tau$-functions. We show that these $\tau$-functions can be
identified with the $\tau$-functions for the Drinfeld--Sokolov hierarchy of
$\mathfrak{g}$-type. Explicit examples for $\mathfrak{g}=A_1$ and $A_2$ are
provided, which are connected to the KdV hierarchy and the Boussinesq hierarchy
respectively.
