## Abstract

We provide examples to extend a recent conjecture concerning the relation between zero curvature representations and nonlocal terms of inverse recursion operators to all recursion operators in dimension two. Namely, we conjecture that nonlocal terms of recursion operators are always related to a suitable zero-curvature representation, not necessarily depending on a parameter or taking values in a semisimple algebra. In particular, the conventional pseudodifferential recursion operators correspond to abelian Lie algebras.

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## References

A. H. Bilge, “On the equivalence of linearization and formal symmetries as integrability tests for evolution equations,”

*J. Phys. A: Math. Gen.*,**26**, 7511–7519 (1993).A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor’kova, I. S. Krasil’shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, and A. M. Vinogradov,

*Symmetries and Conservation Laws for Differential Equations of Mathematical Physics*, Trans. Math. Monogr.,**182**, Amer. Math. Soc., Providence, Rhode Island (1999).M. V. Foursov, “Classification of certain integrable coupled potential KdV and modified KdV-type equations,”

*J. Math. Phys.*,**41**, 6173–6185 (2000).W. Fulton and J. Harris,

*Representation Theory. A First Course*, Springer-Verlag, Berlin (2001).M. Gürses, A. Karasu and V. V. Sokolov, “On construction of recursion operators from Lax representation,”

*J. Math. Phys.*,**40**, 6473–6490 (1999).G. A. Guthrie, “Recursion operators and nonlocal symmetries,”

*Proc. Roy. Soc. London A*,**446**, 107–114 (1994).J. E. Humphreys,

*Introduction to Lie Algebras and Representation Theory*, Springer-Verlag, New York (1972).S. Igonin and R. Martini, “Prolongation structure of the Krichever-Novikov equation,”

*J. Phys. A: Math. Gen.*,**35**, 9801–9810 (2002).A. Karasu (Kalkanlı), A. Karasu, and S. Yu. Sakovich, “A strange recursion operator for a new integrable system of coupled Korteweg-de Vries equations,”

*Acta Appl. Math.*,**83**, 85–94 (2004).I. S. Krasil’shchik and P. H. M. Kersten,

*Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations*, Kluwer, Dordrecht (2000).I. S. Krasil’shchik and P. H. M. Kersten, “Complete integrability of the coupled KdV-mKdV system,” in:

*Lie Groups, Geometric Structures, and Differential Equations. One Hundred Years After Sophus Lie*(T. Morimoto, H. Sato, and K. Yamaguchi, eds.),*Adv. Stud. Pure Math.*,**37**, Math. Soc. Jpn. (2002), pp. 151–171.P. D. Lax, “Periodic solutions of the KdV equation,”

*Commun. Pure Appl. Math.*,**28**, 141–188 (1975).M. Marvan, “Some local properties of Bäcklund relations,”

*Acta Appl. Math.*,**54**, 1–25 (1998).M. Marvan, “Another look on recursion operators,” in:

*Differential Geometry and Applications. Proc. Conf. Brno, 1995*, Masaryk University, Brno (1987), pp. 393–402;*http://www.emis.de/proceedings*.M. Marvan, “Recursion operators for the Einstein equations with symmetries,” in:

*Proc. Conf. “Symmetry in Nonlinear Mathematical Physics,” Kyiv, Ukraine, June 23–28, 2003*, Proc. Inst. Math. NAS Ukraine,**50**, Part I (2004), pp. 179–183.M. Marvan and A. Sergyeyev, “Recursion operator for the Knizhnik-Veselov-Novikov equation,”

*J. Phys. A: Math. Gen.*,**36**, L87–L92 (2003).P. J. Olver, “Evolution equations possessing infinitely many symmetries,”

*J. Math. Phys.*,**18**, 1212–1215 (1977).S. Yu. Sakovich, “Cyclic bases of zero-curvature representations: five illustrations to one concept,”

*Acta Appl. Math.*,**83**, 69–83 (2004).A. Sergyeyev,

*Locality of symmetries generated by nonstandard recursion operators: a new application for implectic-symplectic factorization*, Preprint Math. Inst. Opava, GA 2/2006.A. Sergyeyev, “A strange recursion operator demystified,”

*J. Phys. A: Math.Gen.*,**38**, L257–L262 (2005).A. Sergyeyev and D. Demskoi,

*The Sasa-Satsuma (complex mKdV II) and the complex sine-Gordon II equation revisited: recursion operators, nonlocal symmetries and more*, preprint nlin.SI/0512042.L. A. Takhtadzhyan and L. D. Faddeev,

*Hamiltonian Methods in the Theory of Solitons*, Springer-Verlag, Berlin (1987).V. E. Zakharov and A. B. Shabat, “Integration of nonlinear equations of mathematical physics by the inverse scattering method, II,”

*Funkts. Anal. Prilozh.*,**13**, No. 3, 13–22 (1979).

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Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 23–33, 2006.

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Baran, H., Marvan, M. A conjecture concerning nonlocal terms of recursion operators.
*J Math Sci* **151**, 3083–3090 (2008). https://doi.org/10.1007/s10958-008-9030-6

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DOI: https://doi.org/10.1007/s10958-008-9030-6