On isochronous Bruschi-Ragnisco-Ruijsenaars-Toda lattices: Equilibrium configurations, behaviour in their neighbourhood, diophantine relations and conjectures

Isochronous versions of the Bruschi-Ragnisco-Ruijsenaars-Toda lattice and of some of its, also integrable, variants are introduced, their equilibrium configurations are found (when they exist), and by investigating the motions of these systems near equilibrium some diophantine relations are obtained as well as some insight into the solution of those of these integrable models whose solutions are not yet known.