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We consider analytic differential operators of order $n$ defined on two-dimensional manifolds. Namely, linear operators locally of the form $$\sum_{1 \le i+j \le n} f_{ij} \left(\frac\partial{\partial x}\right)^i \left(\frac\partial{\partial y}\right),$$ with $f_{ij}$ analytic functions. After introducing the notion of elementary singular point for such operators, we discuss a theorem of resolution of singularities, generalizing the classical result of Bendixson-Seidenberg for vector fields in dimension two.