In this paper we exploit further the rich symmetries of the global parameters in transitive mechanical systems. Preliminary results were already published as a research announcement. We show that the main parameters Im, Σ′ (subsets of a Euclidean space) of their Smale's global description depend only on the Lie algebra via the co-adjoint action on its dual, and not on the Lie group itself, solving an old conjecture by the author. We also prove that Im, Σ′ are semialgebraic whenever the adjoint group is algebraic, which in particular holds if the original group is compact, and maybe even semisimple. It is also shown that Re and σ are stratified sets, and the last one is semialgebraic on fibers as it happens to be with the first one. Finally, the notion of Smale's amended potential is generalized for our systems, and some basic properties carry over from the celestial mechanics case, provided that the invariant subsets of the angular momentum are stratified sets.