In 1967, physicist M. Toda exhibited soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. Soon afterwards, the lattice was written in Lax form by Flaschka and was recognized as a completely integrable system with an iso-spectral flow.  In the decades that followed, this system has been generalized in different directions, each with its own analytic, geometric, and topological characteristics. These are known collectively as the Toda lattice. The following collection of papers studies Toda systems from a geometric perspective, highlighting singular flows and non-generic level sets of the constants of motion.  Included are several papers extending facts about group actions, for the purpose of better understanding flows of the Toda lattice. 

Y. Kodama, B.A. Shipman, The finite non-periodic Toda lattice: a geometric and topological viewpoint. To appear in MEMPhys (Modern Encyclopedia of Mathematical Physics) and on Springerlink in 2011 (50 pages). Preprint: http://www.uta.edu/math/preprint/rep2008_03.pdf

B. Shipman, 40 years of the Toda lattice: A topological and geometric viewpoint.  Proceedings of the Workshop on the Iso Level Sets of Integrable Systems 2007, Keio University, MATH COE, Integrative Mathematical Sciences, 2007.  

B. Shipman, The full Kostant -Toda lattice: Geometry of its singularities and its connection to honeybees and physics.  Proceedings of the Workshop on the Iso Level Sets of Integrable Systems 2007, Keio University, MATH COE, Integrative Mathematical Sciences, 2007.

B. A. Shipman, On the Connectedness of a Centralizer. Applied Mathematics Letters 20:467-469, 2007  

B. A. Shipman, Fixed points of unipotent  group actions in Bruhat  cells of a flag manifold. JP Journal of Algebra, Number Theory & Applications 3(2):301-313, 2003 

B. A. Shipman, Compactifiedisospectral  sets of complex tridiagonalHessenberg  matrices. In: Dynamical Systems and Differential Equations, Eds. W. Feng, S. Hu, and X. Lu., American Institute of Mathematical Sciences, p. 788 – 797, 2003 

B. A. Shipman, A unipotent  group action on a flag manifold and "gap sequences" of permutations. Journal of Algebra and Its Applications 2(2):215-222, 2003 

B. A. Shipman, On length and level in the classical Lie algebras. JP Journal of Algebra, Number Theory & Applications 3(1):157-167, 2003

B. A. Shipman, On the fixed-point sets of torus actions on flag manifolds. Journal of Algebra and Its Applications 1(3):1-11, 2002

B. A. Shipman, Nongeneric flows in the full Kostant -Toda lattice. Contemporary Mathematics: Integrable Systems, Topology, and Physics 309:219-249, 2002, American Mathematical Society

B. A. Shipman, On the fixed points of the Toda hierarchy. Contemporary Mathematics: The Geometrical Study of Differential Equations 285:39-49, 2001, American Mathematical Society

B. A. Shipman, The geometry of the full Kostant -Toda lattice ofSl (4,C ). Journal of Geometry and Physics 33:295-325, 2000

B. A. Shipman, A symmetry of order two in the full Kostant -Toda lattice. Journal of Algebra 215:682-693, 1999

B. A. Shipman, Monodromy near the singular level set in theSl (2,C ) Toda lattice. Physics Letters A 239:246-250, 1998

B. A. Shipman, On the geometry of certain isospectral  sets in the full Kostant -Toda lattice. Pacific Journal of Mathematics 181(1):159-186, 1997