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 the steps,
similar to those for the Schr\"{o}dinger 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 non-linear 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 non-singular integrable
potentials (dielectric permitivity) for the Maxwell equations in a 2D
inhomogeneous medium.Comment: 16 pages; keywords Darboux (Moutard) transformation, Classical
acoustic spectral problem, Reflexionless potentials, Soliton
