Classical r\bold{r}-Matrices and Compatible Poisson Structures for Lax Equations on Poisson Algebras

Given a classical $r$-matrix on a Poisson algebra, we show how to construct a natural family of compatible Poisson structures for the Hamiltonian formulation of Lax equations. Examples for which our formalism applies include the Benny hierachy, the dispersionless Toda lattice hierachy, the dispersionless KP and modified KP hierachies, the dispersionless Dym hierachy etc.Comment: 28 page

Spin Calogero-Moser systems associated with simple Lie algebras

We introduce spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. Our analysis is based on a group-theoretic framework similar in spirit to the standard classical $r$-matrix theory for constant $r$-matrices.Comment: 6 page

Integrable spin Calogero-Moser systems

We introduce spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated Lax representations (with spectral parameter) and fundamental Poisson bracket relations. The associated integrable models (called integrable spin Calogero-Moser systems in the paper) and their Lax pairs are then obtained via Poisson reduction and gauge transformations. For Lie algebras of $A_{n}$-type, this new class of integrable systems includes the usual Calogero-Moser systems as subsystems. Our method is guided by a general framework which we develop here using dynamical Lie algebroids.Comment: 30 pages, Latex fil