  We construct a point transformation between two integrable systems, the
multi-component Harry Dym equation and the multi-component extended Harry Dym
equation, that does not preserve the class of multi-phase solutions. As a
consequence we obtain a new type of wave-like solutions, generalising
the~multi-phase solutions of the multi-component extended Harry Dym equation.
Our construction is easily transferable to other integrable systems with
analogous properties.
