  The Burchnall-Chaundy problem is classical in differential algebra, seeking
to describe all commutative subalgebras of a ring of ordinary differential
operators whose coefficients are functions in a given class. It received less
attention when posed in the (first) Weyl algebra, namely for polynomial
coefficients, while the classification of commutative subalgebras of the Weyl
algebra is in itself an important open problem. Centralizers are
maximal-commutative subalgebras, and we review the properties of a basis of the
centralizer of an operator $L$ in normal form, following the approach of K.R.
Goodearl, with the ultimate goal of obtaining such bases by computational
routines. Our first step is to establish the Dixmier test, based on a lemma by
J. Dixmier and the choice of a suitable filtration, to give necessary
conditions for an operator $M$ to be in the centralizer of $L$. Whenever the
centralizer equals the algebra generated by $L$ and $M$, we call $L$, $M$ a
Burchnall-Chaundy (BC) pair. A construction of BC pairs is presented for
operators of order $4$ in the first Weyl algebra. Moreover, for true rank $r$
pairs, by means of differential subresultants, we effectively compute the fiber
of the rank $r$ spectral sheaf over their spectral curve.
