Biharmonic bases in algebras of the second rank (Q1089467)

Let A be an algebra over \({\mathbb{C}}\) of dimension 2, a basis of A, \(\{e_ 1,e_ 2\}\), with nilpotent \(e^ 2_ 1+e^ 2_ 2\neq 0\) is called biharmonic, as the components of Hausdorff differentiable functions from A to A are biharmonic, see the author and \textit{V. F. Kovalev} [Dokl. Akad. Nauk Ukr. SSR, Ser. A. 1981, No.8, 26-27 (1981; Zbl 0472.31001)]. In this paper the algebras are determined which have at least one biharmonic basis, the possible forms of such a basis are discussed.