Abstract
The symmetry constraint for dispersionless Harry Dym (dHD) hierarchy is derived for the first time by taking dispersionless limit of that for 2+1 dimensional Harry Dym hierarchy. Then, the dHD is extended by means of the symmetry constraint which we derived. From the zero-curvature equation of the new extended dHD hierarchy, two types of dHD equations with self-consistent sources (dHDESCS) together with their associated conservation equations are obtained. Moreover, the hodograph solutions to the first type of dHDESCS are given. Finally, Bäcklund transformation between the extended dispersionless mKP hierarchy and extended dHD Hierarchy are also constructed.
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