  Fano orbifold lines are classified by the Dynkin diagrams of type $A,D,$ and
$E$. It is known that the corresponding total descendant potential is a
tau-function of an appropriate Kac--Wakimoto hierarchy. It is also known that
in the A-case the Kac--Wakimoto hierarchies admit an extension and that the
total descendant potential is a tau-function of an extended Kac--Wakimoto
hierarchy. The goal of this paper is to prove that in the D-case the total
descendent potential is also a tau-function of an extended Kac--Wakimoto
hierarchy.
