  In this paper we discuss two canonical transformations that turn St\"{a}ckel
separable Hamiltonians of Benenti type into polynomial form: transformation to
Vi\`ete 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 St\"{a}ckel Hamiltonians of Benenti type. Moreover we
present all geometric ingredients of these Hamiltonians in both Vi\`ete and
Newton coordinates.
