  We consider the classical factorization problem of a third order ordinary
differential operator $L-\lambda$, for a spectral parameter $\lambda$. It is
assumed that $L$ is an algebro-geometric operator, that it has a nontrivial
centralizer, which can be seen as the affine ring of curve, the famous
"spectral curve" $\Gamma$. In this work we explicitly describe the ring
structure 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 a
non-planar spectral curve arises, for an operator of this type. Based on the
structure of the centralizer, we give a symbolic algorithm, using differential
subresultants, to factor $L-\lambda_0$ for all but a finite number of points
$P=(\lambda_0 , \mu_0 , \gamma_0)$ of the spectral curve .
