Algebraic curves as a source of separable multi-Hamiltonian systems

In this paper we systematically consider various ways of generatingintegrable and separable Hamiltonian systems in canonical and in non-canonicalrepresentations from algebraic curves on the plane. In particular, we considerSt\"ackel transform between two pairs of St\"ackel systems, generated by2n-parameter algebraic curves on the plane, as well as Miura maps betweenSt\"ackel systems generated by (n+N)-parameter algebraic curves, leading tomulti-Hamiltonian representation of these systems.