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In this paper we construct first examples of smooth projective surfaces ofgeneral type satisfying the following conditions: there are 1) an ampleintegral 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 theexistence of non-trivial commutative subalgebras of rank one in $\hat{D}$, acompletion of the algebra of partial differential operators in two variables,which can be thought of as a simple algebraic analogue of the algebra ofanalytic pseudodifferential operators on a manifold. We extract theseconditions by elaborating the classification theorem of commutative subalgebrasin $\hat{D}$ due to the second author for the case of rank one subalgebras.Amazingly, the commutative subalgebras with such spectral surfaces do not admitisospectral deformations.

Amazing examples of nonrational smooth spectral surfaces