The conserved polynomials of the Korteweg‒de Vries equation ut=uxxx−12uux are characterized by the vanishing of the residues of their associated differential polynomials evaluated on the formal power series of the kind u=x−2+u0+ Σn≥2unxn.
François Treves. "An algebraic characterization of the Korteweg-de Vries hierarchy." Duke Math. J. 108 (2) 251 - 294, 1 June 2001. https://doi.org/10.1215/S0012-7094-01-10823-5