Classical integrable finite-dimensional systems related to Lie algebras

During the last few years many dynamical systems have been identified, that are completely integrable or even such to allow an explicit solution of the equations of motion. Some of these systems have the form of classical one-dimensional many-body problems with pair interactions; others are more general. All of them are related to Lie algebras, and in all known cases the property of integrability results from the presence of higher (hidden) symmetries. This review presents from a general and universal viewpoint the results obtained in this field during the last few years. Besides it contains some new results both of physical and mathematical interest.The main focus is on the one-dimensional models of n particles interacting pairwise via potentials V(q) = g²ν(q) of the following 5 types: $\text{ν}{}_{\mathrm{<mtext>I</mtext>}}\text{(q)=q}{}^{\mathrm{?2}}\text{, ν}{}_{\mathrm{<mtext>II</mtext>}}\text{(q)=a}{}^{\mathrm{?2}}\text{sinh}{}^2\text{(aq), ν}{}_{\mathrm{<mtext>III</mtext>}}\text{(q)=a}{}^2\text{/}\text{sin}\text{2(aq), ν}{}_{\mathrm{<mtext>IV</mtext>}}\text{=a}{}^2\text{P}\text{(aq), , ν}{}_{\mathrm{<mtext>V</mtext>}}\text{(q)=q}{}^{\mathrm{?2}}\text{+ω}{}^2\text{q}{}^2$. Here $\text{P}$(q) is the Weierstrass function, so that the first 3 cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbor potential, g_j²exp-a(q_j?q_j+1)], is moreover considered. Various generalizations of these models, naturally suggested by their association with Lie algebras, are also treated.