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We consider the classical factorization problem of a third order ordinarydifferential operator $L-\lambda$, for a spectral parameter $\lambda$. It isassumed that $L$ is an algebro-geometric operator, that it has a nontrivialcentralizer, which can be seen as the affine ring of curve, the famous"spectral curve" $\Gamma$. In this work we explicitly describe the ringstructure of the centralizer of $L$ and, as a consequence, we prove that$\Gamma$ is a space curve. In this context, the first computed example of anon-planar spectral curve arises, for an operator of this type. Based on thestructure of the centralizer, we give a symbolic algorithm, using differentialsubresultants, to factor $L-\lambda_0$ for all but a finite number of points$P=(\lambda_0 , \mu_0 , \gamma_0)$ of the spectral curve .

Factoring Third Order Ordinary Differential Operators over Spectral Curves