On classification of some surfaces of revolution of finite type

In this article, we study the following problem of [5]: Classify all finite type surfaces in a Euclidean 3-space E3. A surface M in a Euclidean 3-space is said to be of finite type if each of its coordinate functions is a finite sum of eigenfunctions of the Laplacian operator on M with respect to the induced metric (cf. [1,2]). Minimal surface are the simplest examples of surfaces of finite type, in fact, minimal surfaces are of l-type. The spheres, minimal surfaces and circular cylinders are the only known exampls of surfaces of finite type in E3 and it seems to be the only finite type surfaces in E3 (cf. [5]). The first author conjectured in [2] that spheres are the only compact finite type surfaces in E3. Since then, it was prived step by sted and separately that finite type tubes, finite type ruled surfaces, finite type quadrics and finite type cones are surfaces of the only known examples (cf. [2,6,7,10].) Our next natural target for this classification problem is the class of surfaces of revolution. However, this case seems to be much difficult than the other cases mentioned above. We therefore investigate this classification problem for this class and obtain classification theorems for surfaces of revolution which are either of rational or of polynomial kinds (cf. §1 for the definitions). As consequence, further supports for the conjecture cited above are obtained