Pantelis Damianou

We construct a symplectic realization and a bi-Hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Plücker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.

We construct a large family of Hamiltonian systems which interpolate between the classical Kostant–Toda lattice and the full Kostant–Toda lattice and we discuss their integrability. There is one such system for every nilpotent ideal $\mathcal{I}$ in a Borel subalgebra $\mathfrak{b}_+$ of an arbitrary simple Lie algebra $\mathfrak{g}$. The classical Kostant–Toda lattice corresponds to the case of $\mathcal{I}=[\mathfrak{n}_+, \mathfrak{n}_+]$, where $\mathfrak{n}_+$ is the unipotent ideal of $\mathfrak{b}_+$, while the full Kostant–Toda lattice corresponds to $\mathcal{I}=\{0\}$. We mainly focus on the case $\mathcal{I}=[[\mathfrak{n}_+, \mathfrak{n}_+], \mathfrak{n}_+]$. In this case, using the theory of root systems of simple Lie algebras, we compute the rank of the underlying Poisson manifolds and construct a set of (rational) functions in involution, large enough to ensure Liouville integrability. These functions are restrictions of well-chosen integrals of the full Kostant–Toda lattice, except for the case of the Lie algebras of type $C$ and $D$ where a different function (Casimir) is needed. The latter fact, and other ones listed in the paper, point at the Liouville integrability of all the systems we construct, but also at the nontriviality of obtaining the result in full generality.

We examine a class of Lotka–Volterra equations in three dimensions which satisfy the Kowalevski–Painlevé property. We restrict our attention to Lotka–Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painlevé analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions. We also show that the conditions are sufficient.

We construct a new symplectic, bi-Hamiltonian realization of the KM-system by reducing the corresponding one for the Toda lattice. The bi-Hamiltonian pair is constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper we also review the important work of Moser on the Toda and KM-systems.

We establish, using a new approach, the integrability of a particular case in the Kozlov–Treshchev classification of Birkhoff integrable Hamiltonian systems. The technique used is a modification of the so called quadratic Lax pair for $D_n$ Toda lattice combined with a method used by M. Ranada in proving the integrability of the Sklyanin case.

Results on the finite nonperiodic $A_n$ Toda lattice are extended to the Bogoyavlensky–Toda systems of type $B_n$ and $C_n$. The investigated areas include master symmetries, recursion operators, higher Poisson brackets and invariants. A conjecture which relates the degrees of higher Poisson brackets and the exponents of the corresponding Lie group is verified for these systems.