The Darboux process and a noncommutative bispectral problem

The Darboux process, also known by many other names, played a very important role in some extremely enjoyable joint work that Hans and I did 25 years ago. I revisit a version of this problem in a case when scalars are replaced by matrices, i.e., elements of a non-commutative ring. Many of the issues studied here can be pushed to the case of a ring with identity, but my emphasis is on very concrete examples involving $2\times 2$ matrices.