Commutative subalgebras of the ring of differential operators on a curve (Q583292)

Can an affine algebraic curve X over \({\mathbb{C}}\) be recovered from its algebra of differential operators \({\mathcal D}(X)?\) Yes, if the affine curve is not simply connected, for, as is shown, precisely in that case the affine ring of X equals the set of the locally ad-nilpotent elements of \({\mathcal D}(X)\). If X is simply connected, in general there exists a maximal commutative subalgebra of \({\mathcal D}(X)\), consisting of locally ad-nilpotent elements, which is not isomorphic to the affine ring.