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 Stäckel transform between two pairs of Stäckel systems, generated by 2n-parameter algebraic curves on the plane, as well as Miura maps between Stäckel systems generated by (n+N)-parameter algebraic curves, leading to multi-Hamiltonian representation of these systems.

Comment: Version accepted for publication in Special Issue 2 of Open Communications in Nonlinear Mathematical Physics. One reference added, some language issues fixed, additional explanations in Introduction

• 14H70 - Relationships between algebraic curves and integrable systems
• 37J35 - Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
• 70H20 - Hamilton-Jacobi equations in mechanics