The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward B？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？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 Painlevé III equation in trigonometric form is defined by a classical Darboux transform of a harmonic inverse mean curvature surface of revolution. 1. Introduction The theory of surfaces is connected with the theory of solitons through a compatibility condition of the Gauss-Weingarten equations. Bobenko [1] gave an outline of eight classes of surfaces in the three-dimensional Euclidean space in the formulation of the theory of solitons. They are minimal surfaces, surfaces of constant mean curvature, surfaces of constant positive Gaussian curvature, surfaces of constant negative Gaussian curvature, Bonnet surfaces, harmonic inverse mean curvature surfaces, Bianchi surfaces, and Bianchi surfaces of positive curvature. For the investigation of these surfaces, a matrices representation of quaternions is used to write their moving frames. Their moving frames are integrated by Sym's formula [2]. Quaternionic analysis by Pedit and Pinkall [3] is a technology to investigate surfaces in the Euclidean three- or four-space which are related to the soliton theory. In this theory, the Euclidean four-space is modeled on the set of all quaternions . A quaternionic line trivial bundle with a complex structure over a Riemann surface is associated with a conformal map from the Riemann surface to . We can assume that a quaternionic line trivial bundle associated with a constrained Willmore surface in equips a harmonic complex structure [4]. If a constrained Willmore surface is neither minimal nor superconformal, then this complex structure defines a smooth family of flat connections on the line bundle. Then a holonomy spectral curve of a constrained Willmore torus is defined by a smooth family of holonomies of . The relation between a constrained Willmore torus and its holonomy spectral curve is discussed in detail in [4]. If a conformal map from a torus to is not a constrained Willmore torus, then the quaternionic line trivial bundle associated with the conformal map is accompanied with a nonharmonic complex structure. For a conformal map, its Darboux transform is
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