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We address the classical factorization problem of a one dimensionalSchr\"odinger operator $-\partial^2+u-\lambda$, for a stationary potential $u$of the KdV hierarchy but, in this occasion, a "parameter" $\lambda$. Inspiredby the more effective approach of Gesztesy and Holden to the "direct" spectralproblem, we give a symbolic algorithm by means of differential eliminationtools to achieve the aimed factorization. Differential resultants are used forcomputing spectral curves, and differential subresultants to obtain the firstorder common factor. To make our method fully effective, we design a symbolicalgorithm to compute the integration constants of the KdV hierarchy, in thecase of KdV potentials that become rational under a Hamiltonian change ofvariable. Explicit computations are carried for Schr\"odinger operators withsolitonic potentials.

Factorization of KdV Schr\"odinger operators using differential subresultants