  In this paper we study in detail algebraic properties of the algebra
$\mathcal D(W)$ of differential operators associated to a matrix weight of
Gegenbauer type. We prove that two second order operators generate the algebra,
indeed $\mathcal D(W)$ is isomorphic to the free algebra generated by two
elements subject to certain relations. Also, the center is isomorphic to the
affine algebra of a singular rational curve. The algebra $\mathcal D(W)$ is a
finitely-generated torsion-free module over its center, but it is not flat and
therefore it is not projective.
  This is the second detailed study of an algebra $\mathcal D(W)$ and the first
one coming from spherical functions and group representations. We prove that
the algebras for different Gegenbauer weights and the algebras studied
previously, related to Hermite weights, are isomorphic to each other. We give
some general results that allow us to regard the algebra $\mathcal D(W)$ as the
centralizer of its center in the Weyl algebra. We do believe that this should
hold for any irreducible weight and the case considered in this paper
represents a good step in this direction.
