On divisors of small canonical degree on Godeaux surfaces

Pre-spectral data $(X,C,D)$ coding the rank-1 commutative subalgebras of a certain completion $\widehat D$ of the algebra of differential operators $D=k[[x_1,x_2]][\partial_1,\partial_2]$, where $k$ is an algebraically closed field of characteristic 0, are shown to exist. Here $X$ is a Godeaux surface, $C$ is an effective ample divisor represented by a smooth curve, $h^0(X,\mathscr O_X(C))=1$ and $D$ is a divisor on $X$ satisfying the conditions $(D, C)_X=g(C)-1$, $h^i(X,\mathscr O_X(D))=0$ for $i=0,1,2$ and $h^0(X,\mathscr O_X(D+C))=1$.
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