  The Matrix Bochner Problem aims to classify which weight matrices have their
sequence of orthogonal polynomials as eigenfunctions of a second-order
differential operator. Casper and Yakimov, in [2], demonstrated that, under
certain hypotheses, all solutions to the Matrix Bochner Problem are
noncommutative bispectral Darboux transformations of a direct sum of classical
scalar weights. This paper aims to provide the first proof that there are
solutions to the Matrix Bochner Problem that do not arise through a
noncommutative bispectral Darboux transformation of any direct sum of classical
scalar weights. This initial example could contribute to a more comprehensive
understanding of the general solution to the Matrix Bochner Problem.
