ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2011, Vol. 32, No. 1, pp. 61–70. c  Pleiades Publishing, Ltd., 2011. Properties of the Calogero–Degasperis–Ibragimov–Shabat Differential Sequence M. Euler * , N. Euler ** , and P.G.L. Leach 1*** (Submitted by M.A. Malakhaltsev) Department of Mathematics, Lule ˚ a University of Technology SE-971 87 Lule ˚ a, Sweden 1 School of Mathematics, University of KwaZulu-Natal, Private Bag X54001 Durban 4000, Republic of South Africa Received September 9, 2010 Abstract—We present a differential sequence based upon the Calogero–Degasperis–Ibragimov– Shabat Equation and determine first integrals and the general solution. Under suitable transforma- tions each member of the differential sequence can be recast as a product of two factors and we report some of the properties of the factored form. DOI: 10.1134/S1995080211010070 Keywords and phrases: Differential sequence; factored form; singularity analysis; symmetry analysis. 1. INTRODUCTION The evolution equation known as the Calogero–Degasperis–Ibragimov–Shabat equation is u t = u 3x +3u 2 u xx +9uu 2 x +3u 4 u x . (1.1) The equation is integrable and can be linearised by means of a nonlocal transformation [4, 13, 14]. Equation (1.1) can be written in the potential form v t = v 3x − 3 4 v 2 xx v x +3v x v xx + v 3 x , (1.2) where v x = u 2 (1.3) v t =2uu xx − u 2 x +6u 3 u x + u 6 . (1.4) In [13] it was shown that (1.1) possesses a recursion operator of albeit rather formidable appearance. Consequently there exists an hierarchy of evolution equations generated from (1.1) by repeated applica- tion of this recursion operator. Euler and Euler [9] demonstrated that there exist potential forms for the elements of the hierarchy and that the second potential form was linear. Corresponding to an hierarchy of evolution partial differential equations generated by a recursion operator one can also define a sequence of ordinary differential equations generated by the same operator. This was illustrated in the case of sequences beginning with the Riccati and Ermakov-Pinney equations [7]. In the case of the Riccati Differential Sequence a detailed study of the singularity and symmetry properties is found in [2]. Furthemore alternate sequences were introduced [8], which hold the key to the first integrals of the proper sequences and to identify integrable sequences. * E-mail: marianna@sm.luth.se ** E-mail: norbert@sm.luth.se *** E-mail: leachp@ukzn.ac.za;leach@math.aegean.gr 61