The evolution equations of several exactly solvable dynamical systems can be written in Hamiltonian form in two distinct manners, using two different Poisson bracket structures and corresponding Hamiltonians. Such bi Hamiltonian structures lead to the existence of conserved quantities associated with the integrability of the pertinent systems. In this talk we review our results on bi-Hamiltonian structures of integrable many-body models of point particles moving along one dimension, which are also coupled to internal `spin’ degrees of freedom. The models of our interest belong to the celebrated family of Calogero–Moser–Sutherland and Toda type systems. They will be viewed as shadows (alias Poisson reductions) of simple higher dimensional bi-Hamiltonian systems having large symmetry groups.