On isochronous Shabat-Yamilov-Toda lattices: Equilibrium configurations, behavior in their neighborhood, diophantine relations and conjectures

Isochronous variants are introduced of two integrable lattices introduced by A.B. Shabat and R.I. Yamilov, their equilibrium configurations are found (in the cases when they exist), and by investigating the motions of these systems near equilibrium some diophantine relations are discovered.