A Strange Recursion Operator Demystified

We show that a new integrable two-component system of KdV type studied by Karasu (Kalkanlı) et al. (arXiv: nlin.SI/0203036) is bihamiltonian, and its recursion operator, which has a highly unusual structure of nonlocal terms, can be written as a ratio of two compatible Hamiltonian operators found by us. Using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in invo-lution, and the evolution systems corresponding to these symmetries are bihamiltonian as well. We also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form, with the integration operator D −1 appearing in each term at most once. Using the Panilevé test, Karasu (Kalkanlı) [1] and Sakovich [2] found a new integrable evolution system of KdV type, u t = 4u xxx − v xxx − 12uu x + vu x + 2uv x , v t = 9u xxx − 2v xxx − 12vu x − 6uv x + 4vv x , (1) and a zero curvature representation for it [2]. Notice that this system is, up to a linear transformation of u and v, equivalent to the system (16) from the Foursov's [3] list of two-component evolution systems of KdV type possessing (homogeneous) symmetries of order k, 4 ≤ k ≤ 9. Karasu (Kalkanlı), Karasu and Sakovich [4] found that (1) has a recursion operator of the form