Infinitely many local generalized symmetries without recursion operator or master symmetry: integrability of the Foursov–Burgers system revisited

where an α is a real parameter. For α = 0 the system (1) is equivalent [1] to an equation found by Svinolupov [2] while for α = 1 this system is equivalent to the system (4.13) in Olver and Sokolov [3]. Finally, for α = 1/2 a recursion operator for Eq.(1) was found in [1]. Thus, Eq.(1) is known to be integrable for α = 0, 1/2, 1. For α 6= 0, 1/2, 1 the system (1) turned out to have [1] six generalized symmetries but no recursion operator or master symmetry was found so far, and hence it was not known whether Eq.(1) is integrable in any reasonable sense and, in particular, whether it admits infinitely many generalized symmetries. In view of the recent results of Sanders and van der Kamp [4] who exhibited several examples of two-component triangular evolution systems that possess only a finite number (greater than one) of local generalized symmetries, it is natural to ask whether Eq.(1) could provide an example of a non-triangular system with finitely many local generalized symmetries. In the present paper we show that this is not the case: the system (1) has infinitely many commuting local generalized symmetries and, what is more, the system in question is C-integrable, i.e., its general solution can be found. Quite unusual, however, is the fact that these symmetries are generated using a nonlocal two-term recursion relation (8) rather than a recursion operator, see Theorem 3 below for details. Moreover,