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

Commuting Ordinary Differential Operators and the Dixmier Test