Seminarium Nieliniowość i Geometria

A system of real ordinary differential equations (ODE), $\ddot{y}_k=\sum_{l=1}^n M_{kl}\exp y_l,~k=1,...,n$, where $M$ is a real $n\times n$ constant symmetric matrix, is discussed in detail for general $n$. After a symmetry analysis and a discussion of the Lagrangian-Hamiltonian structure, a one-parameter family of exact solutions is found and proved to be unstable to small perturbations. Next, the system is tested for the Painlev\'e property. The result shows that the only completely integrable cases of this system are equivalent to the Toda lattice by similarity transformations of matrix $M$. A few physical applications are discussed for the general and some special $n$'s.