With T. Nevins. **Flows of Calogero-Moser Systems**.

The Calogero-Moser (or CM) particle system and its generalizations
appear, in a variety of ways, in integrable systems, nonlinear PDE,
representation theory, and string theory. Moreover, the partially
completed CM systems--in which dynamics of particles are continued
through collisions--have been identified as meromorphic Hitchin
systems, giving natural ``geometric action-angle variables'' for the
CM system. Motivated by relations of the CM system to nonlinear PDE,
we introduce a new class of generalizations of the spin CM particle
systems, the framed (rational, trigonometric and elliptic) CM
systems. We give two algebro-geometric descriptions of these
systems, via meromorphic Hitchin systems with decorations (framing
data) on (cuspidal, nodal and smooth) cubic curves and via
one-dimensional sheaves on corresponding ``twisted'' ruled surfaces.
We also present a simple geometric formulation of the flows of all
meromorphic GL_n Hitchin systems (with no regularity assumptions) as
tweaking flows on spectral sheaves. Using this formulation, we show
that all spin and framed CM systems are identified with hierarchies
of tweaking flows on the corresponding spectral sheaves. This
generalizes the well-known description of spinless CM systems in
terms of tangential covers.