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We study the classical Rajeev-Ranken model, a Hamiltonian system with threedegrees of freedom describing nonlinear continuous waves in a 1+1-dimensionalnilpotent scalar field theory pseudodual to the SU(2) principal chiral model.While it loosely resembles the Neumann and Kirchhoff models, its equations maybe viewed as the Euler equations for a centrally extended Euclidean algebra.The model has a Lax pair and r-matrix leading to four generically independentconserved quantities in involution, two of which are Casimirs. Their level setsdefine four-dimensional symplectic leaves on which the system is Liouvilleintegrable. On each of these leaves, the common level sets of the remainingconserved quantities are shown, in general, to be 2-tori. The non-generic levelsets can only be horn tori, circles and points. They correspond to measure zerosubsets where the conserved quantities develop relations and solutionsdegenerate from elliptic to hyperbolic, circular or constant functions. Ageometric construction allows us to realize each common level set as a bundlewith base determined by the roots of a cubic polynomial. A dynamics is definedon the union of each type of level set, with the corresponding phase manifoldsexpressed as bundles over spaces of conserved quantities. Interestingly,topological transitions in energy hypersurfaces are found to occur at energiescorresponding to horn tori, which support purely homoclinic orbits. Thedynamics on each horn torus is non-Hamiltonian, but expressed as a gradientflow. Finally, we discover a family of action-angle variables for the systemthat apply away from horn tori.

Invariant tori, action-angle variables and phase space structure of the Rajeev-Ranken model