  In this paper we study the isomorphism classes of local, Artinian, Gorenstein
k-algebras A whose maximal ideal M satisfies dim_k(M^3/M^4)=1 by means of
Macaulay's inverse system generalizing a recent result by J. Elias and M.E.
Rossi. Then we use such results in order to complete the description of the
singular locus of the Gorenstein locus of the punctual Hilbert scheme of degree
11.
