It is shown that the Poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one-form in three dimensions. Advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability. Techniques for finding Poisson structures are presented and applied to various examples such as the Halphen system which has been studied as the two-monopole problem by Atiyah and Hitchin. It is shown that the Halphen system can be formulated in terms of a flat SL(2,R)-valued connection and belongs to a nontrivial Godbillon-Vey class. On the other hand, for the Euler top and a special case of three-species Lotka-Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi-Hamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the SL(2,R) structure is a quadratic unfolding of an integrable one-form in 3 + 1 dimensions. It is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of Poisson structure and some new techniques for incorporating arbitrary constants into the Poisson one-form are presented herein. This leads to some extensions, analogous to q extensions, of Poisson structure. The Kermack-McKendrick model and some of its generalizations describing the spread of epidemics, as well as the integrable cases of the Lorenz, Lotka-Volterra, May-Leonard, and Maxwell-Bloch systems admit globally integrable bi-Hamiltonian structure.

Exact solutions of the Deser-Jackiw-Templeton field equations including a cosmological constant are presented. They generalize the finite-action homogeneous, anisotropic vacuum solutions of topologically massive gravity. We find that only Vuorio-type solutions, where any two of the constant scale factors coincide, admit a cosmological constant. One of these solutions can be dressed up to yield a two parameter black hole solution of topologically massive gravity.

Hamiltonian methods are used to study spherically symmetric gravitational fields with electromagnetism and a massless scalar field as sources. The constraints reduce to a first-order linear ordinary differential equation which is solved to yield the dynamical Hamiltonian, or the complete solution when no dynamics is present in the problem.

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The bi-Hamiltonian structure for a large class of one-dimensional hyberbolic systems of conservation laws in two field variables, including the equations of gas dynamics, shallow water waves, one-dimensional elastic media, and the Bom-Infeld equation from nonlinear electrodynamics, is exhibited. For polytropic gas dynamics, these results lead to a quadri-Hamiltonian structure. New higher-order entropy-flux pairs (conservation laws) and higherorder symmetries are exhibited.