  A tau function of the 2D Toda hierarchy can be obtained from a generating
function of the two-partition cubic Hodge integrals. The associated Lax
operators turn out to satisfy an algebraic relation. This algebraic relation
can be used to identify a reduced system of the 2D Toda hierarchy that emerges
when the parameter $\tau$ of the cubic Hodge integrals takes a special value.
Integrable hierarchies of the Volterra type are shown to be such reduced
systems. They can be derived for positive rational values of $\tau$. In
particular, the discrete series $\tau = 1,2,\ldots$ correspond to the Volterra
lattice and its hungry generalizations. This provides a new explanation to the
integrable structures of the cubic Hodge integrals observed by Dubrovin et al.
in the perspectives of tau-symmetric integrable Hamiltonian PDEs.
