  In a companion paper to this one, we proved that the Gromov--Witten theory of
a Fano orbifold line of type $D$ is governed by a system of Hirota Bilinear
Equations. The goal of this paper is to prove that every solution to the Hirota
Bilinear Equations determines a solution to a new integrable hierarchy of Lax
equations. We suggest the name extended D-Toda hierarchy for this new system of
Lax equations, because it should be viewed as the analogue of Carlet's extended
bi-graded Toda hierarchy, which is known to govern the Gromov--Witten theory of
Fano orbifold lines of type $A$
