  In this paper we systematically consider various ways of generating
integrable and separable Hamiltonian systems in canonical and in non-canonical
representations from algebraic curves on the plane. In particular, we consider
St\"ackel transform between two pairs of St\"ackel systems, generated by
2n-parameter algebraic curves on the plane, as well as Miura maps between
St\"ackel systems generated by (n+N)-parameter algebraic curves, leading to
multi-Hamiltonian representation of these systems.
