Direct summands of syzygy modules of the residue class field

Let R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the n-th syzygy of the residue field of R has a nontrivial direct summand of finite G-dimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely.