The even and odd supersymmetric Hunter–Saxton and Liouville equations

It is shown that two different supersymmetric extensions of the Harry Dym equation lead to two different negative hierarchies of the supersymmetric integrable equations. While the first one yields the known even supersymmetric Hunter–Saxton equation, the second one is a new odd supersymmetric Hunter–Saxton equation. It is further proved that these two supersymmetric extensions of the Hunter–Saxton equation are reciprocally transformed to two different supersymmetric extensions of the Liouville equation.