  The string equation of type $(2,2g+1)$ may be thought of as a higher order
analogue of the first Painlev\'e equation that corresponds to the case of $g =
1$. For $g > 1$, this equation is accompanied with a finite set of commuting
isomonodromic deformations, and they altogether form a hierarchy called the PI
hierarchy. This hierarchy gives an isomonodromic analogue of the well known
Mumford system. The Hamiltonian structure of the Lax equations can be
formulated by the same Poisson structure as the Mumford system. A set of
Darboux coordinates, which have been used for the Mumford system, can be
introduced in this hierarchy as well. The equations of motion in these Darboux
coordinates turn out to take a Hamiltonian form, but the Hamiltonians are
different from the Hamiltonians of the Lax equations (except for the lowest one
that corresponds to the string equation itself).
