  The problem of finding weight matrices $W(x)$ of size $N \times N$ such that
the associated sequence of matrix-valued orthogonal polynomials are
eigenfunctions of a second-order matrix differential operator is known as the
Matrix Bochner Problem.
  This paper aims to study Darboux transformations between these weight
matrices and to establish a direct connection with the structure of the algebra
$\mathcal D(W)$. We find several general properties and provide an explicit
description of the Darboux equivalence classes for scalar classical weights.
Additionally, we determine the algebra $\mathcal{D}(W)$ when $W$ is a direct
sum of classical scalar weights.
