H Sadeghi - Mathematics Department, Vali-e-Asr University, Rafsanjan, Iran.
S.M Hoseini - Mathematics Department, Vali-e-Asr University, Rafsanjan, Iran.
S. Eskandar - Mathematics Department, Vali-e-Asr University, Rafsanjan, Iran.

Using Lax representation, hereditary operator related to the Harry-Dym ( HD ) hierarchy is constructed. Also Lie analysis and Lax pair are used to drive the conserved quantities for the HD hierarchy. Lie symmetry method is also applied to study Painlev´e test for the family of HD hierarchy. It is found that for this family the Painlev´e property can be at most sufficient for integrability, but not necessary. This fact has been proved earlier for just first member of this family.

Lie analysis, conservation laws, hereditary operator, Painlev´e property, ARS conjecture