  The notion of a generalized harmonic inverse mean curvature surface in the
Euclidean four-space is introduced. A backward B\"{a}cklund transform of a
generalized harmonic inverse mean curvature surface is defined. A Darboux
transform of a generalized harmonic inverse mean curvature surface is
constructed by a backward B\"{a}cklund transform. For a given isothermic
harmonic inverse mean curvature surface, its classical Darboux transform is a
harmonic inverse mean curvature surface. Then a transform of a solution to the
Painlev\'{e} III equation in trigonometric form is defined by a classical
Darboux transform of a harmonic inverse mean curvature surface of revolution.
