On the overdetermined system about surfaces with parallel mean curvature vector field. (Q1872583)

The author studies two-dimensional surfaces embedded in a two-dimensional complex space form. Due to \textit{T. Ogata} [see Kodai Math. J. 18, No. 3, 397--407 (1995; Zbl 0865.53007)] the property that such an immersion has a parallel mean curvature vector field is equivalent to the existence of a certain overdetermined system \(S\) of differential equations. All solutions of the system \(S\) were given by \textit{K. Kenmotsu} and \textit{D. Zhou} [Am. J. Math. 122, 295--317 (2000; Zbl 0997.53006)]. In the present paper the author constructs the solutions of \(S\) by a different method which also uses the theory of polynomials.