  The Burchnall-Chaundy theory concerns the classification of all pairs of
commuting ordinary differential operators. We phrase this theory in the
language of spectral data for integrable systems.
  In particular, we define spectral data for rank 1 commutative algebras $A$ of
ordinary differential operators. We solve the inverse problem for such data,
i.e. we prove that the algebra $A$ is (essentially) uniquely determined by its
spectral data. The isomorphy type of $A$ is uniquely determined by the
underlying spectral curve.
