In this article, we present an explicit linearization of dynamical systems of Ruijsenaars-Schneider (RS) type and of the perturbations introduced by F Calogero [2] of these systems with all orbits periodic of the same period. The existence of this linearization and its algebraic nature relies on the dynamical equation firstly discussed in the article [3]. The notion of algebraic linearization which was first displayed in NEEDS 99 conference will be discussed further with several other examples in a forthcoming publication. A differential system is algebraically (resp. analytically) linearizable if there are n globally defined functions (rational, resp. meromorphic) which are generically independent so that the time evolution of the flow expressed in these functions is linear (in time) and algebraic in the initial coordinates.