On the dispersionless Dym hierarchy of hydrodynamic type: two approaches

Based on Kodama-Gibbons approach we study hydrodynamic-type N reductions of the dispersionless Dym hierarchy. In particular, for N-truncated Lax operators, the hodograph coefficients associated with hodograph equations can be expressed in simple formulae. On the other hand, we also use Takasaki-Takebe approach to rederive the hodograph coefficients in twistor construction and discuss special solutions from deformations of twistor data. Our result elucidates the equivalence of the two approaches.