## Abstract

For a rational differential operator \({L=AB^{-1}}\), the Lenard–Magri scheme of integrability is a sequence of functions \({F_n, n \geq 0}\), such that (1) \({B(F_{n+1})=A(F_n)}\) for all \({n \geq 0}\) and (2) the functions \({B(F_n)}\) pairwise commute. We show that, assuming that property (1) holds and that the set of differential orders of \({B(F_n)}\) is unbounded, property (2) holds if and only if *L* belongs to a class of rational operators that we call integrable. If we assume moreover that the rational operator *L* is weakly non-local and preserves a certain splitting of the algebra of functions into even and odd parts, we show that one can always find such a sequence (*F*
_{
n
}) starting from any function in Ker *B*. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations.

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## Additional information

Communicated by: Yasuyuki Kawahigashi

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Carpentier, S. A sufficient condition for a rational differential operator to generate an integrable system.
*Jpn. J. Math.* **12, **33–89 (2017). https://doi.org/10.1007/s11537-016-1619-9

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### Keywords and phrases

- integrable systems
- Lenard–Magri scheme of integrability
- rational pseudo-differential operators
- symmetries

### Mathematics Subject Classification (2010)

- 37K10
- 17B80
- 35Q53
- 35S05
- 37K10