Rigidity of Kac-Schwarz operators

In his work on the mathematical formulation of 2d quantum gravity Schwarz established a rigidity result for Kac-Schwarz operators for the n-KdV hierarchies. Later on, Adler and van Moerbeke as well as Fastr'{e} obtained different proofs of this result. We give yet another proof of the rigidity, one that in fact holds for all Drinfeld-Sokolov hierarchies.