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We study the deformed Harry Dym and Hunter-Zheng equations with two arbitrarydeformation parameters. These reduce to various other known models inappropriate limits. We show that both these systems are bi-Hamiltonian with thesame Hamiltonian structures. They are integrable and belong to the samehierarchy corresponding to positive and negative flows. We present the Lax pairdescription for both the systems and construct the conserved charges ofnegative order from the Lax operator. For the deformed Harry Dym equation, weconstruct the non-standard Lax representation for two special classes of valuesof the deformation parameters. In general, we argue that a non-standarddescription will involve a pseudo-differential operator of infinite order.Comment: Latex file, 15 page

Deformed Harry Dym and Hunter–Zheng equations