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We determine all the potentials V(x) for the Schrόdinger equation (dl + Y(x))φ = k 2 φ such that some family of eigenfunctions φ satisfies a differential equation in the spectral parameter k of the formFor each such V{x) we determine the algebra of all possible operators B and the corresponding functions Θ(x). Table of Contents 0. Introduction 177 1. (adL) m + 1 ((9) = 0 180 2. K(oo) is Finite 187 3. The Rational KdV Potentials 192 4. The Even Case 203 5. The Even Potentials Work Too 213 6. F(oo) = oo is the Airy Case 218 7. Some Illustrative Examples 222

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