Surprising examples of nonrational smooth spectral surfaces

In this paper we study necessary and sufficient algebro-geometric conditions for the existence of a nontrivial commutative subalgebra of rank $1$ in $\widehat{D}$, a completion of the algebra of partial differential operators in two variables, which can be thought of as a simple algebraic analogue of the algebra of analytic pseudodifferential operators on a manifold.
These are conditions on a projective (spectral) surface; they are encoded in a new notion of pre-spectral data. For smooth surfaces the sufficient conditions look especially simple. On a smooth projective surface there should exist 1) an ample integral curve $C$ with $C^2=1$ and $h^0(X,\mathscr{O}_X(C))=1$; 2) a divisor $D$ with $(D, C)_X=g(C)-1$, $h^i(X,\mathscr{O}_X(D))=0$, $i=0,1,2$, and $h^0(X,\mathscr{O}_X(D+C))=1$. Amazingly, there are examples of such surfaces for which the corresponding commutative subalgebras do not admit isospectral deformations.
Bibliography: 45 titles.