On the Lie-Algebraic Integrability of the Calogero-Degasperis Dynamical System and Its Generalizations

Abstract

We studied the Lax type integrability of the Calogero-Degasperis nonlinear dynamical system, possessing only one local conserved quantity. Based on the gradientholonomic integrability approach there are stated tboth the bi-Hamiltonian structure of the Calogero-Degasperis dynamical system and isomorphism of its symmetries group to the semidirect product of the diffeomorphism group of the circle and the abelian group of functions on it. We also constructed a rich algebra of non-Hamiltonian symmetries, related to the Bäcklund transformed general symmetries of the corresponding linearization of the Calogero-Degasperis dynamical system. There is also analyzed in detail the inverse problem of classifying integrable generalized Calogero-Degasperis type dynamical systems a priori possessing a finite number of conserved quantities.