  We consider local non-Gorenstein rings of the form $(S_i,\mathfrak{n}_i)=k[X,
Y_1, \ldots ,Y_i]/\left(X^2, (Y_1, \ldots, Y_i)^2\right), $ where $i\geq 2.$ We
show that every totally reflexive $S_i$-module has a presentation matrix of the
form $I x + \sum_{j=1}^i B_j y_j, $ where $I$ is the identity matrix and $B_j$
is an square matrix with entries in the residue field, $k$. From there, we
prove that there exists a bijection between the set of isomorphism classes of
totally reflexive modules (without projective summands) over $S_i$ which are
minimal generated by $n$ elements and the set of $i$-tuples of $n \times n$
matrices with entries in $k$ modulo a certain equivalence relation.
