In this paper we discuss two canonical transformations that turn Stäckel separable Hamiltonians of Benenti type into polynomial form: transformation to Viète coordinates and transformation to Newton coordinates. Transformation to Newton coordinates has been applied to these systems only very recently and in this paper we present a new proof that this transformation indeed leads to polynomial form of Stäckel Hamiltonians of Benenti type. Moreover we present all geometric ingredients of these Hamiltonians in both Viète and Newton coordinates.
Funding: The research of J.D. Maniraguha and C. Kurujyibwami was supported by International Science Programme (ISP, Uppsala University) in collaboration with Eastern Africa Universities Mathematics Programme (EAUMP). The research of K. Marciniak was partially supported by the Swedish International Development Cooperation Agency (Sida) through the Rwanda- Sweden bilateral research cooperation.