ABUNDANT SYMMETRIES FOR THE 1+1 DIMENSIONAL CLASSICAL LIOUVILLE FIELD-THEORY

Explicit calculation shows us that there exists a rich associated symmetry structure and abundant strong symmetry configuration for the 1 + 1 dimensional classical Liouville field theory. Starting from these symmetries and strong symmetries, various integrable hierarchies can be obtained. The Liouville I-III hierarchies, the modified Korteweg-de Vries (KdV) hierarchy, the Fordy-Gibbons I, and Fordy-Gibbons II hierarchies are just special cases while the KdV hierarchy, Caudry-Dodd-Gibbon-Sawada-Kotera hierarchy, and Kaup-Kupershmidt hierarchy can be obtained by using a same Miura transformation from the modified KdV, the Fordy-Gibbons I and Fordy-Gibbons II hierarchies, respectively. Two types of the generalized Riccati hierarchies are also presented.