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We provide a complete list of two- and three-component Poisson structures ofhydrodynamic type with degenerate metric, and study their homogeneousdeformations. In the non-degenerate case any such deformation is trivial, thatis, can be obtained via Miura transformation. We demonstrate that in thedegenerate case this class of deformations is non-trivial, and depends on acertain number of arbitrary functions. This shows that the secondPoisson-Lichnerowicz cohomology group does not vanish.

On deformations of one-dimensional Poisson structures of hydrodynamic type with degenerate metric