Amazing examples of nonrational smooth spectral surfaces

In this paper we construct first examples of smooth projective surfaces of general type satisfying the following conditions: there are 1) an ample integral curve $C$ with $C^2=1$ and $h^0(X,O_X(C))=1$; \quad 2) a divisor $D$ with $(D, C)_X=g(C)-1$, $h^i(X,O_X(D))=0$, $i=0,1,2$, and $h^0(X, O_X(D+C))=1$. Such conditions arise from necessary and sufficient conditions for the existence of non-trivial commutative subalgebras of rank one in $\hat{D}$, 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. We extract these conditions by elaborating the classification theorem of commutative subalgebras in $\hat{D}$ due to the second author for the case of rank one subalgebras. Amazingly, the commutative subalgebras with such spectral surfaces do not admit isospectral deformations.