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.