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We study discrete isospectral symmetries for the classical
acoustic spectral problem in spatial dimensions one and two by
developing a Darboux (Moutard) transformation formalism for this
problem. The procedure follows steps similar to those for the
Schrödinger operator. However, there is no one-to-one
correspondence between the two problems. The technique developed
enables one to construct new families of integrable potentials for
the acoustic problem, in addition to those already known. The
acoustic problem produces a nonlinear Harry Dym PDE. Using the
technique, we reproduce a pair of simple soliton solutions of this
equation. These solutions are further used to construct a new
positon solution for this PDE. Furthermore, using the
dressing-chain approach, we build a modified Harry Dym equation
together with its LA pair. As an application, we construct some
singular and nonsingular integrable potentials (dielectric
permitivity) for the Maxwell equations in a 2D inhomogeneous
medium
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