The Complete Set of Generalized Symmetries for the Calogero–Degasperis–Ibragimov–Shabat Equation

All known today integrable scalar (1+1)-dimensional evolution equations with time-independent coefficients possess infinite-dimensional Abelian algebras of time-independent higher order symmetries (see e.g. [1, 2]). However, the equations of this kind usually do not have local timedependent higher order symmetries. The only known exceptions from this rule seem to occur [3] for linearizable equations like e.g. the Burgers equation, for which the complete set of symmetries was found in [4]. In the present paper we confirm this for a third order linearizable equation (4), which is referred below as Calogero–Degasperis–Ibragimov–Shabat equation, and exhibit the complete set of its time-dependent local generalized symmetries. This equation was discovered by Calogero and Degasperis [5] and studied, among others, by Ibragimov and Shabat [6], Svinolupov and Sokolov [7], Sokolov and Shabat [8], Calogero [9], and by Sanders and Wang [10]. The paper is organized as follows. In Section 2 we recall some well known definitions and results on the symmetries of evolution equations. In Section 3 we present the main result – Theorem 1, giving the complete description of the set of all local generalized symmetries for CDIS equation.